Dasatinib and Venetoclax in Treating Patients With Philadelphia Chromosome Positive or BCR-ABL1 Positive Early Chronic Phase Chronic Myelogenous Leukemia

Inclusion Criteria: * Diagnosis of Ph-positive or BCR-ABL positive CML in early chronic phase CML (i.e., time from diagnosis ≤ 12 months). Except for hydroxyurea and/or 1 to 2 doses of cytarabine patients must have received no or minimal prior therapy, defined as \<1 month (30 days) of prior FDA approved TKI. * Clonal evolution defined as the presence of additional chromosomal abnormalities other than the Ph chromosome has historically been included as a criterion for accelerated phase. However, patients with clonal evolution as the only criterion of accelerated phase have a significantly better prognosis, and when present at diagnosis may not impact the prognosis at all. Thus, patients with clonal evolution and no other criteria for accelerated phase will be eligible for this study. * Eastern Cooperative Oncology Group (ECOG) performance of 0-2 * Total bilirubin \< 1.5 x upper limit normal (ULN) (unless secondary to Gilbert's disease, in which case should be \< 2.5x ULN) * Serum glutamate pyruvate transaminase (SGPT) \< 3 x ULN * Creatinine \< 1.5 x ULN * Patients must sign an informed consent indicating they are aware of the investigational nature of this study, in keeping with the policies of the hospital Exclusion Criteria: * New York Heart Association (NYHA) cardiac class 3-4 heart disease * Patients meeting the following criteria are not eligible unless cleared by cardiology: * Uncontrolled angina within 3 months * Diagnosed or suspected congenital long QT syndrome * Any history of clinically significant ventricular arrhythmias (such as ventricular tachycardia, ventricular fibrillation, or Torsades de pointes) * Prolonged QTc interval on pre-entry electrocardiogram (\> 460 msec) * History of significant bleeding disorder unrelated to cancer, including: * Diagnosed congenital bleeding disorders (e.g., von Willebrand's disease) * Diagnosed acquired bleeding disorder within one year (e.g., acquired anti-factor VIII antibodies) * Patients with active, uncontrolled psychiatric disorders include: psychosis, major depression, and bipolar disorders * Subject is known to be positive for human immunodeficiency virus (HIV); (HIV testing is not required) * Evidence of other clinically significant uncontrolled condition(s) including, but not limited to: * Uncontrolled and/or active systemic infection (viral, bacterial or fungal) * Chronic hepatitis B virus (HBV) or hepatitis C (HCV) requiring treatment. Note: subjects with serologic evidence of prior vaccination to HBV (i.e. hepatitis B surface (HBs) antigen negative-, anti-HBs antibody positive and anti-hepatitis B core (HBc) antibody negative) or positive anti-HBc antibody from intravenous immunoglobulins (IVIG) may participate. * Women of pregnancy potential must practice an effective method of birth control during the course of the study, in a manner such that risk of failure is minimized. Prior to study enrollment, women of childbearing potential (WOCBP) must be advised of the importance of avoiding pregnancy during trial participation and the potential risk factors for an unintentional pregnancy. Postmenopausal women must be amenorrheic for at least 12 months to be considered of non-childbearing potential. Women must continue birth control for the duration of the trial and at least 3 months after the last dose of study drug. Pregnant or breast-feeding women are excluded. All WOCBP must have a negative pregnancy test prior to first receiving investigational product. If the pregnancy test is positive, the patient must not receive study drug and must not be enrolled in the study. * Patients in late chronic phase (i.e., time from diagnosis to treatment \> 12 months), accelerated or blast phase are excluded; the definitions of CML phases are as follows: * Early chronic phase: * Time from diagnosis to therapy ≤12 months * Late chronic phase: * Time from diagnosis to therapy \> 12 months * Blastic phase: * Presence of 30% blasts or more in the peripheral blood or bone marrow * Accelerated phase CML: * Presence of any of the following features: * Peripheral or marrow blasts 15% or more * Peripheral or marrow basophils 20% or more * Thrombocytopenia \< 100 x 10\^9/L unrelated to therapy * Documented extramedullary blastic disease outside liver or spleen