Venetoclax and Quizartinib in Treating Patients With FLT3-mutated Recurrent or Refractory Acute Myeloid Leukemia

Inclusion Criteria: * FLT3-ITD mutated patients with relapsed/refractory AML (up to four prior therapeutic regimens for AML i.e. up to salvage 4 AML), including patients who may have been previously exposed to prior FLT3-inhibitor/s other than quizartinib (stem cell transplant \[SCT\] or stem cell therapy for patients who previously underwent SCT/stem cell therapy in remission will not be considered a salvage regimen) * Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 2 * Serum direct bilirubin =\< 1.5 x upper limit normal (ULN) (or =\< 3.0 x ULN if deemed to be elevated due to leukemia) * Alanine aminotransferase and/or aspartate aminotransferase (aspartate transaminase) =\< 2.5 x ULN (or =\< 5.0 x ULN if deemed elevated due to leukemia) * Subjects with documented Gilbert's Syndrome may have a total bilirubin \> 1.5 x ULN * Potassium levels should be within institutional normal limits * Magnesium levels should be within institutional normal limits * Calcium (normalized for albumin) levels should be within institutional normal limits * Adequate renal function as demonstrated by a serum creatinine =\< 1.8 * Patients must provide written informed consent * With the exception of patients with rapidly proliferative disease, the interval from prior treatment to time of initiation of venetoclax and quizartinib administration will be at least 14 days or at least 5 half-lives (whichever is shorter) for cytotoxic/noncytotoxic agents. The half-life for the therapy in question will be based on published pharmacokinetic literature (abstracts, manuscripts, investigator brochures, or drug-administration manuals) and will be documented in the protocol eligibility document. The use of chemotherapeutic or anti-leukemic agents is not permitted during the study with the following exceptions: * Intrathecal (IT) therapy for patients with controlled central nervous system (CNS) leukemia at the discretion of the principal investigator (PI). Controlled CNS leukemia is defined by the absence of active clinical signs of CNS disease and no evidence of CNS leukemia on the most recent 2 simultaneous cerebrospinal fluid (CSF) evaluations * Use of one dose of cytarabine (up to 2 g/m\^2) or hydroxyurea for patients with rapidly proliferative disease is allowed before the start of study therapy and on therapy. These medications will be recorded in the case-report form * Baseline ejection fraction by echocardiogram (ECHO) or multigated acquisition scan (MUGA) must be \>= 50% * Women of non-childbearing potential are those who are postmenopausal greater than 1 year or who have had a bilateral tubal ligation or hysterectomy * Women of childbearing potential must agree to have a negative serum or beta human chorionic gonadotropin (beta-hCG) pregnancy test result within 7 days prior to the first dose of study drugs and must agree to use an effective contraception method during the study and for 90 days following the last dose of the study. Men who have partners of childbearing potential must agree to use an effective contraceptive method during the study and for 90 days following the last dose of study drug Exclusion Criteria: * Subject has t(8;21) or inv(16) karyotype abnormalities * Subject has acute promyelocytic leukemia (French-American-British Class M3 AML) * Prior exposure to quizartinib at any time in the past * Serum potassium \< 3.5 mEq/L despite supplementation, or \> 5.5 mEq/L. Serum magnesium above or below the institutional normal limit despite adequate management. Serum calcium (corrected for albumin levels) above or below institutional normal limit despite adequate management * Patients with known allergy or hypersensitivity to quizartinib, venetoclax or any of their components * Subject with a known history of being human immunodeficiency virus (HIV) positive (due to potential drug-drug interactions between antiretroviral medications and venetoclax, as well as anticipated venetoclax mechanism-based lymphopenia that may potentially increase the risk of opportunistic infections) * Note: HIV testing is not required * Subject has consumed grapefruit, grapefruit products, Seville oranges (including marmalade containing Seville oranges) or star fruit within 3 days prior to the initiation of study treatment * Subject has a significant history of renal, neurologic, psychiatric, endocrinologic, metabolic, immunologic, hepatic, cardiovascular disease, or any other medical condition that in the opinion of the investigator and/or the PI would adversely affect his/her participating in this study. Patients who have had any major surgical procedure within 14 days of day 1 * Subject has a malabsorption syndrome or other condition that precludes enteral route of administration * Subject exhibits evidence of other clinically significant uncontrolled condition(s) including, but not limited to: a. Uncontrolled systemic infection requiring intravenous (IV) therapy (viral, bacterial or fungal). Infections controlled on concurrent anti-microbial agents are acceptable, and anti-microbial prophylaxis per institutional guidelines is acceptable. Patients with neutropenic fever considered infection related should be afebrile for at least 72 hours prior to first dose * Subject has a history of other malignancies within 1 year prior to study entry, with the exception of: * Adequately treated in situ carcinoma of the cervix uteri or carcinoma in situ of breast * Basal cell carcinoma of the skin or localized squamous cell carcinoma of the skin * Previous malignancy confined and surgically resected (or treated with other modalities) with curative intent * Patients on active antineoplastic or radiation therapy for a concurrent malignancy at the time of screening. Maintenance therapy, hormonal therapy, or steroid therapy for well-controlled malignancy is allowed * Patients with a known positive hepatitis B or C infection by serology, with the exception of those with an undetectable viral load within 3 months (hepatitis B or C testing is not required prior to study entry). Subjects with serologic evidence of prior vaccination to hepatitis B virus (HBV) (i.e., hepatitis B surface antigen negative \[HBs Ag-\], and hepatitis B surface antibody positive \[anti-HBs+\]) may participate * Female subjects who are pregnant or breastfeeding * Impaired cardiac function including any of the following: * Screening electrocardiogram (ECG) with a corrected QT (QTc) \> 450 msec. The QTc interval will be calculated by Fridericia's correction factor (QTcF) at screening and on day 1 prior to the first dose of quizartinib. The QTcF will be derived from the average QTcF in triplicate. If QTcF \> 450 msec on day 1, quizartinib will not be given. * Patients with congenital long QT syndrome * History or presence of sustained ventricular tachycardia requiring medical intervention within 3 months prior to starting study drug * Any history of clinically significant ventricular fibrillation or torsades de pointes * Known history of second or third degree heart block (may be eligible if the patient currently has a pacemaker) within 3 months prior to starting study drug * Sustained heart rate of \< 50/minute on screening or day 1 ECG * Right bundle branch block + left anterior hemiblock (i.e. bifascicular block) * Isolated right bundle branch block (RBBB) will not be an exclusion criterion * Complete left bundle branch block * Patients with myocardial infarction or unstable angina within 6 months prior to starting study drug * Congestive heart failure (CHF) New York (NY) Heart Association class III or IV within 3 months prior to starting study drug * Atrial fibrillation documented within 2 weeks prior to first dose of study drug * Patients who are actively taking CYP3A inducers. CYP3A4 inducers should be stopped at least 3 days prior to the first dose of quizartinib and are prohibited at any time on study. Moderate and strong CYP3A4 inhibitors should be stopped at least 3 days prior to the first dose of quizartinib and are prohibited during cycle 1. Moderate (but not strong) CYP3A4 inhibitors may be used with the below dose reductions of venetoclax after cycle 1. Patients may receive weak CYP3A4 inhibitors at any time on study. The venetoclax and quizartinib doses do not need to be adjusted for weak CYP3A4 inhibitors * Patients who require treatment with concomitant drugs that prolong QT/QTc interval. QT/QTc prolonging drugs should be stopped at least 3 days prior to the first dose of quizartinib and are prohibited at any time on study * Known family history of congenital long QT syndrome