Testing the Addition of Navitoclax to the Combination of Dabrafenib and Trametinib in People Who Have BRAF Mutant Melanoma

Inclusion Criteria: * PHASE I SUBJECTS ONLY: Prior therapy is allowed; for patients enrolled in the Phase I portion of the study, patients may have received any number of prior lines of therapy including treatment with a BRAF and/or MEK inhibitor; prior navitoclax use will not be allowed, unless the patient received \< 7 days of navitoclax lead-in on this or another study and had to stop for reasons other than toxicity or disease progression * Patients must have histologically confirmed, BRAF-mutant (V600E/K) melanoma (molecularly confirmed using validated, commercially available assay performed in a Clinical Laboratory Improvement Act \[CLIA\]-approved laboratory) that is metastatic or unresectable and for which standard curative measures do not exist or are no longer effective * If test at CLIA-certified lab used a non-Food and Drug Administration (FDA) approved method, information about the assay must be provided; (FDA approved tests for BRAF V600 mutations in melanoma include: THxID BRAF Detection Kit and Cobas 4800 BRAF V600 Mutation Test) * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral CT scan, MRI, or calipers by clinical exam * Prior therapy is allowed; for patients enrolled in the Phase II portion of the study, patients may have received prior immunotherapy (including high-dose IL-2, ipilimumab, nivolumab, and other anti-PD1/PDL1 antibodies) or chemotherapy; however prior navitoclax, BRAF inhibitor and/or MEK inhibitor therapy will not be allowed * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of navitoclax in combination with dabrafenib and trametinib in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%) * Life expectancy of greater than 3 months * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1 x 10\^9/L * Hemoglobin \>= 9 g/dl (patients may be transfused to this level) * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 x institutional upper limit of normal OR \> 1.5 x institutional upper limit of normal allowed if direct bilirubin is within normal range * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Prothrombin time (PT)/international normalized ratio (INR) and partial thromboplastin time (PTT) \< 1.3 x upper limit of normal (ULN) * Serum creatinine =\< 1.5 mg/dL OR creatinine clearance \>= 50 mL/min/1.73 m\^2 * Left ventricular ejection fraction \>= institutional lower limit of normal (LLN) by ECHO * Patients must have a corrected QT (QTc) interval of less than 480 msec * The effects of navitoclax, dabrafenib, and trametinib on the developing human fetus are unknown; for this reason, women of child-bearing potential and men with partners of childbearing potential must agree to use adequate contraception (barrier method of birth control, or abstinence; hormonal contraception is not allowed due to drug-drug interactions which can render hormonal contraceptives ineffective) prior to study entry, for the duration of study participation, and for 4 months after completion of study drug administration; should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately; based on studies in animals, it is also known that dabrafenib may cause damage to the tissue that makes sperm; this may cause sperm to be abnormal in shape and size and could lead to infertility, which may be irreversible; safety and efficacy of the combination of dabrafenib and trametinib in pediatric populations have not been investigated; dabrafenib or trametinib-dabrafenib combination should not be administered to pediatric populations outside clinical trials * Able to swallow and retain oral medication and must not have any clinically significant gastrointestinal abnormalities that may alter absorption such as malabsorption syndrome or major resection of the stomach or bowels * Ability to understand and the willingness to sign a written informed consent document; if a patient has impaired decision-making capacity, a legally authorized representative, patients will be allowed to participate Exclusion Criteria: * PHASE I SUBJECTS ONLY: Patients must not have received prior navitoclax, unless the patient received \< 7 days of navitoclax lead-in on this or another study and had to stop for reasons other than toxicity or disease progression * Patients who have had immunotherapy, chemotherapy or radiotherapy within 14 days prior to the first dose of navitoclax, or prior systemic anti-cancer therapy (chemotherapy with delayed toxicity, extensive radiation therapy, immunotherapy, biologic therapy, or vaccine therapy) within the last 3 weeks prior to first dose of dabrafenib and/or trametinib; chemotherapy regimens without delayed toxicity within the last 2 weeks preceding the first dose of study treatment; biologics will not be allowed within 30 days prior to, or during, navitoclax administration * Prior navitoclax, BRAF inhibitor, and MEK inhibitor is prohibited; (exceptions for Phase I are described above) * Patients who are receiving any other investigational agents have received any other investigational drugs within 28 days (or five half-lives, whichever is shorter; with a minimum of 14 days from the last dose) preceding the first dose of study treatment and during the study * Patients with treated leptomeningeal or brain metastasis are not eligible unless there is demonstrated stability (documented by imaging) for \>= 3 months from any prior treatment of leptomeningeal or brain metastasis. Treatment may include surgery, radiation or systemic therapy. Patients with untreated leptomeningeal or brain metastasis or requiring corticosteroids are not eligible. Subjects on a stable dose of corticosteroids \> 1 month or who have been off of corticosteroids for at least 2 weeks can be enrolled with approval of the Cancer Therapy Evaluation Program (CTEP) medical monitor. Subjects must also be off of enzyme-inducing anticonvulsants for \> 4 weeks * History of allergic reactions attributed to compounds of similar chemical or biologic composition to navitoclax, dabrafenib, or trametinib, or excipients or to dimethyl sulfoxide (DMSO) * Uncontrolled intercurrent illness including, but not limited to, ongoing or active serious infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, uncontrolled diabetes, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because navitoclax, dabrafenib, and trametinib may have teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with the study drugs, breastfeeding should be discontinued if the mother is treated with the study drugs * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy that predict to interact with any of the study drugs are ineligible because of the potential for pharmacokinetic interactions with the study drugs; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated; it is not necessary to conduct HIV testing at screening; patients who are HIV-positive with undetectable viral loads, not on interacting antiretroviral therapy, and have CD4 counts above 300/mm\^3 may be eligible after discussion with the principal investigator * History of another malignancy; exception: patients who have been disease-free for 3 years (depending upon tumor type studied or clinical setting, 3 or 5 years can be used; e.g., for advanced melanoma and pancreatic studies 3 years is more appropriate due to aggressiveness of the disease, while 5 years can be more appropriate for prostate or ovarian cancer or adjuvant setting when life expectancy is longer), or patients with a history of completely resected non-melanoma skin cancer and/or patients with indolent secondary malignancies, are eligible; consult the CTEP medical monitor if unsure whether second malignancies meet the requirements specified above; exception: patients with history of RAS mutation-positive tumors are not eligible regardless of interval from the current study; prospective RAS testing is not required; however, if the results of previous RAS testing are known, they must be used in assessing eligibility * History of interstitial lung disease or pneumonitis * History or current evidence/risk of retinal vein occlusion (RVO) or central serous retinopathy (CSR): * History of RVO or CSR, or predisposing factors to RVO or CSR (e.g., uncontrolled glaucoma or ocular hypertension, uncontrolled systemic disease such as hypertension, diabetes mellitus, or history of hyperviscosity or hypercoagulability syndromes) * Visible retinal pathology as assessed by ophthalmic exam that is considered a risk factor for RVO or CSR such as evidence of new optic disc cupping, evidence of new visual field defects, and intraocular pressure \> 21 mmHg * History or evidence of cardiovascular risk including any of the following: * A QT interval corrected for heart rate using the Bazett's formula QTcB \>= 480 msec on screening electrocardiography (ECG) * History or evidence of current clinically significant uncontrolled arrhythmias (exception: patients with controlled atrial fibrillation for \> 30 days prior to randomization are eligible) * History of acute coronary syndromes (including myocardial infarction and unstable angina), coronary angioplasty, or stenting within 6 months prior to randomization * History or evidence of current \>= class II congestive heart failure as defined by the New York Heart Association (NYHA) functional classification system * Treatment-refractory hypertension defined as a blood pressure of systolic \> 140 mmHg and/or diastolic \> 90 mmHg which cannot be controlled by anti-hypertensive therapy * Abnormal cardiac valve morphology (\>= grade 2) documented by echocardiogram (subjects with grade 1 abnormalities \[i.e., mild regurgitation/stenosis\] can be entered on study); subjects with moderate valvular thickening should not be entered on study * Known history of hepatitis B virus (HBV), or hepatitis C virus (HCV) infection (patients with chronic or cleared HBV and HCV infection are eligible); it is not necessary to conduct HBV and HCV testing at screening * Subject has an underlying condition predisposing them to bleeding or currently exhibits signs of clinically significant bleeding * Subject has a recent history of non-chemotherapy-induced thrombocytopenic-associated bleeding within 1 year prior to the first dose of study drug * A history of glucose-6-phosphate dehydrogenase (G6PD) deficiency * Unresolved toxicity of National Cancer Institute Common Terminology Criteria for Adverse Events, version 5.0 (NCI CTCAE v. 5.0) grade 2 or higher from previous anti-cancer therapy, except alopecia or an endocrine toxicity related to immunotherapy (e.g. thyroiditis/hypothyroidism, adrenal insufficiency, hypophysitis) requiring replacement therapy, at the time of randomization * Due to the expected dose-limiting toxicity of thrombocytopenia, the following concomitant medications are not allowed during navitoclax administration: clopidogrel, ibuprofen, tirofiban, warfarin, and other anticoagulants, drugs, or herbal supplements that affect platelet function are excluded, with the exception of low-dose anticoagulation medications (such as heparin) that are used to maintain the patency of a central intravenous catheter; aspirin will not be allowed within 7 days prior to the first dose of navitoclax or during navitoclax administration; however, subjects who have previously received aspirin therapy for thrombosis prevention may resume a low dose (i.e., maximum 100 mg QD) of aspirin if platelet counts are stable (\>= 50,000/mm\^3) through 6 weeks of navitoclax administration; all decisions regarding treatment with aspirin therapy will be determined by the investigator in conjunction with the medical monitor * Current use of a prohibited medication; the following medications or non-drug therapies are prohibited: * Other anti-cancer therapy while on study treatment; (note: megestrol \[Megace\] if used as an appetite stimulant is allowed) * Concurrent treatment with bisphosphonates is permitted; however, treatment must be initiated prior to the first dose of study therapy; prophylactic use of bisphosphonates in patients without bone disease is not permitted, except for the treatment of osteoporosis * Because the composition, pharmacokinetics (PK), and metabolism of many herbal supplements are unknown, the concurrent use of all herbal supplements is prohibited during the study (including, but not limited to, St. John's wort, kava, ephedra \[ma huang\], ginkgo biloba, dehydroepiandrosterone \[DHEA\], yohimbe, saw palmetto, or ginseng) * Anticoagulants and antiplatelet agents are excluded except for low-dose, aspirin * Preclinical studies indicate that navitoclax is metabolized by CYP3A4, is a moderate inhibitor of CYP2C8, and is a strong inhibitor of CYP2C9; there is also evidence of interactions with dabrafenib; therefore, caution should be exercised when dosing navitoclax concurrently with CYP2C8 and CYP2C9 substrates; common CYP2C8 substrates include paclitaxel, statins, and glitazones, whereas CYP2C9 substrates include phenytoin; when possible, investigators should switch to alternative medications or monitor the patients closely; CYP3A inhibitors such as ketoconazole and clarithromycin are not allowed 7 days prior to the first dose of navitoclax or during navitoclax administration * Patients receiving any medications or substances that are strong inhibitors or inducers of CYP3A or CYP2C8 are ineligible; current use of, or intended ongoing treatment with: herbal remedies (e.g., St. John's wort), or strong inhibitors or inducers of permeability-glycoprotein (Pgp) or breast cancer resistance protein 1 (Bcrp1) should also be excluded; below are a few examples of the agents: * Strong inducers of CYP3A or CYP2C8, since concentrations of dabrafenib may be decreased: * Antibiotics: rifamycin class agents (e.g., rifampin, rifabutin, rifapentine) * Anticonvulsants: carbamazepine, oxcarbazepine, phenobarbital, phenytoin, s-mephenytoin * Miscellaneous: bosentan, St. John's wort * Strong inhibitors of CYP3A or CYP2C8 since concentrations of dabrafenib may be increased: * Antibiotics: clarithromycin, telithromycin, troleandomycin * Antidepressants: nefazodone * Antifungals: itraconazole, ketoconazole, posaconazole, voriconazole * Hyperlipidemia: gemfibrozil * Antiretroviral: ritonavir, saquinavir, atazanavir * Miscellaneous: conivaptan * Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated list; medical reference texts such as the Physicians' Desk Reference may also provide this information; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product