Trabectedin and Venetoclax in Treating Patients With Chronic Lymphocytic Leukemia or Small Lymphocytic Lymphoma Resistant or Intolerant to a BTK Inhibitor

Inclusion Criteria: * Patients with a diagnosis of CLL/SLL who are progressing (based on 2018 International Workshop on Chronic Lymphocytic Leukemia \[iwCLL\] criteria) on or intolerant to a BTK inhibitor (BTK-inhibitor-intolerant is defined as unable to maintain on a stable and continuous dose of at least ibrutinib 140 mg/day \[or acalabrutinib 100 mg/day\] for at least 2 weeks due to recurrent treatment-related grade 2 or higher non-hematologic toxicity by Common Terminology Criteria for Adverse Events \[CTCAE\] grading) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Total bilirubin =\< 1.0 x upper limit of normal (ULN) or =\< 3 x ULN for patients with Gilbert's disease * Creatinine clearance \> 50 mL/min (calculated according to institutional standards or using Cockcroft-Gault formula) * Alanine aminotransferase (ALT), aspartate aminotransferase (AST) =\< 2.5 x ULN * Alkaline phosphatase (ALP) =\< 2.5 x ULN * Platelet (PLT) count \>= 50,000/l, with no platelet transfusion in 2 weeks prior to registration, unless cytopenia is due to bone marrow involvement with CLL, in which case PLT count \> 30,000/l, with no PLT transfusion in 2 weeks prior to registration * Absolute neutrophil count (ANC) \>= 1000/l, unless cytopenia is due to bone marrow involvement with CLL, in which case ANC \> 500/l * Creatine phosphokinase (CPK) \< 2.5 x ULN * Left ventricular ejection fraction (LVEF) assessed by multi-gated acquisition scan (MUGA) or echocardiogram within limits of normal range * Women of childbearing potential must agree to use an effective contraception method during the study and for 60 days following the last dose of study drug. Women of non- childbearing potential are those who are postmenopausal greater than 1 year or who have had a bilateral tubal ligation or hysterectomy. Men who have partners of childbearing potential must agree to use an effective contraceptive method during the study and for 60 days following the last dose of study drug * Free of prior malignancies for 2 years with exception of patients diagnosed with basal cell or squamous cell carcinoma of the skin, or carcinoma "in situ" of the cervix or breast, who are eligible even if they are currently treated or have been treated and/or diagnosed in the past 2 years prior to study enrollment. If patients have another malignancy that was treated within the last 2 years, such patients may be enrolled, if the likelihood of requiring systemic therapy for this other malignancy within 2 years is less than 20% * Patients or their legally authorized representative must provide written informed consent Exclusion Criteria: * Radiotherapy or chemotherapy within 2 weeks prior to the first dose of the study drugs. Given the quick progression associated with resistance to BTK inhibitors, no limits will be placed to the use of BTK inhibitors for enrollment or initiation of treatment on this trial * Uncontrolled active systemic infection (viral, bacterial, and fungal) * Active, uncontrolled autoimmune phenomenon (autoimmune hemolytic anemia or immune thrombocytopenia) requiring steroid therapy with \> 20 mg daily of prednisone or equivalent * Significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, congestive heart failure, or myocardial infarction within 2 months of screening, or any class 3 or 4 cardiac disease as defined by the New York Heart Association Functional Classification * Patient is pregnant or breast-feeding * Venetoclax-refractory CLL or prior treatment with trabectedin * Malabsorption syndrome or other condition that precludes oral/enteral route of administration * Patients who received the following within 7 days prior to first dose of venetoclax: moderate and strong CYP3A inhibitors and inducers, P-glycoprotein inhibitors, or narrow therapeutic index P-glycoprotein substrates AND patients who received the following within 3 days prior to first dose of venetoclax: grapefruit or grapefruit products, Seville oranges and star fruit