A Dose-Finding Phase Ib Study of the Oral BCL-2 Inhibitor Venetoclax (ABT-199) in Combination With Standard Induction Therapy, Dasatinib, Prednisone, (and Rituximab in CD20+ Patients) in Adult Patients With Newly Diagnosed and Relapsed Philadelphia Chromosome Positive ALL (Ph+ ALL) and Ph+ MPAL

• Subjects must have diagnosis of B acute lymphoblastic leukemia harboring the t(9;22) translocation (Philadelphia chromosome positive acute lymphoblastic leukemia \[Ph+ ALL\]) or Ph+ mixed phenotype acute leukemia (MPAL)) confirmed according to criteria.

• Diagnosis of MPAL will only be considered for enrollment during the dose finding period. Individuals with MPAL will not receive per protocol consolidation

• All individuals must have a bone marrow biopsy completed during the screening period. Patients with central nervous system (CNS) disease will be included

• Expression of CD19 by flow cytometry on bone marrow, if individual has received prior chimeric antigen receptor (CAR) T therapy. If CD19 negative, enrollment may be considered for induction treatment only

• Newly diagnosed subjects must have received no prior treatment for their ALL with the exception of steroids (prednisone, dexamethasone), hydrea or IT methotrexate. Individuals may receive pre-treatment with steroids during the screening phase prior to enrollment

• Individuals with relapsed disease may not have had prior treatment with dasatinib, however treatment with other tyrosine kinases is permitted

• At least 3 half-lives must have passed before cycle 1 day 1 (C1D1) for participants that have used strong or moderate CYP3A inducers (such as rifampin, carbamazepine, phenytoin, and St. John's wort) or warfarin or CYP3A inhibitors (such as fluconazole, ketoconazole and clarithromycin) prior to enrollment

• At least 4 weeks must have passed before cycle 1 day 3 (C1D3) for participants that have recently received a live vaccine

• Age \>= 18 years. All participants irrespective of their gender identity and members of all races and ethnic groups will be included

• Eastern Cooperative Oncology Group (ECOG) status =\< 2

• Must be able to take oral medication

• Aspartate aminotransferase (AST) \< 2.5 x upper limit of normal (ULN)

‣ Unless considered due to leukemic organ involvement

• Alanine aminotransferase (ALT) \< 2.5 x ULN

‣ Unless considered due to leukemic involvement

• Total bilirubin \< 1.5 x ULN

‣ Unless considered due to leukemic organ involvement

⁃ Note: subjects with Gilbert's Syndrome may have a bilirubin \> 1.5 x ULN per discussion between the investigator and AbbVie medical monitor

• Subject must have adequate renal function as demonstrated by a calculated creatinine clearance \>= 50 mL/min; determined via urine collection for 24-hour creatinine clearance or by the Cockcroft-Gault formula

• Persons of childbearing potential must have a negative serum or urine pregnancy test (sensitivity \< 25 IU human chorionic gonadotropin \[HCG\]/L) within 72 hours prior to the start of the study drug

• Persons of reproductive potential must agree to use a highly effective method of contraception throughout treatment and for at least 4 weeks after study drug is stopped. Persons of childbearing potential and persons with a sexual partner of childbearing potential must be advised of the importance of avoiding pregnancy during trial participation and the potential risk factors for an unintentional pregnancy

• Normal corrected QT Formula (QTcF) interval on screening electrocardiogram (EKG) (\< 450 ms regardless of sex)

• Ability to understand and the willingness to sign a written informed consent document