Inclusion Criteria:  Subjects must have histologically confirmed diagnosis of pre-B acute lymphoblastic leukemia harboring the t(9;22) translocation (Philadelphia chromosome positive acute lymphoblastic leukemia [Ph+ ALL]). All patients must have a bone marrow biopsy completed during the screening period. Patients with central nervous system (CNS) disease will be included Newly diagnosed subjects must have received no prior treatment for their ALL with the exception of steroids (prednisone, dexamethasone), hydrea or IT methotrexate. Patients may receive up to 6 days of pre-treatment with steroids prior to enrollment during the screening phase Patients with relapsed disease may not have had prior treatment with dasatinib Age >= 18 years. Both men and women 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  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  NOTE: For subjects that have body mass index (BMI) of > 25 kg/m^2, 24-hour measured creatinine clearance is required Women 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 an adequate method of contraception throughout treatment and for at least 4 weeks after study drug is stopped. Women of childbearing potential and men 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 (QTc) interval on screening electrocardiogram (EKG) (< 450 ms in men, < 470 ms in women) Ability to understand and the willingness to sign a written informed consent document  Exclusion Criteria:  For newly diagnosed subjects: who have received treatment with cytotoxic chemotherapy, radiotherapy or immunotherapy for their ALL, or prior dasatinib treatment. For relapsed subjects: prior dasatinib treatment Subjects who have received any investigational agents or subjects who are taking investigational or commercial agents or therapies with the intent to treat the subject's malignancy within seven days of enrollment Subjects with chronic myelogenous leukemia (CML) in myeloid blast crisis, Ph+ acute myeloid leukemia (AML) or acute leukemia of ambiguous lineage Subjects with clinically serious infections as determined by the provider requiring ongoing antibiotic therapy. This does not include antibiotic treatment for neutropenic fever Patients with a pleural or pericardial effusion of any grade Subjects with a history of allergic reactions attributed to compounds of similar chemical or biologic composition to dasatinib or other agents used in the study Subjects who have undergone prior allogeneic hematopoietic stem cell transplantation Subject has received the following within 7 days prior to the initiation of study treatment: Strong or moderate CYP3A inducers (such as rifampin, carbamazepine, phenytoin, and St. John's wort); warfarin or inhibitors (such as fluconazole, ketoconazole and clarithromycin Subjects with uncontrolled cardiac illness including but not limited to, symptomatic congestive heart failure, unstable angina pectoris, clinically significant ventricular arrhythmias (such as ventricular tachycardia, ventricular fibrillation, or Torsades de pointes), or pulmonary hypertension Subjects with diagnosed congenital prolonged QT syndrome Pregnant women are excluded from this study because dasatinib is a pregnancy category D agent with the potential for 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 dasatinib, breastfeeding should be discontinued if the mother is treated with dasatinib. These potential risks may also apply to venetoclax for which the pregnancy category and risks to the fetus are unknown  Participant is seropositive with human immunodeficiency virus (HIV) or has active infection with hepatitis B virus (HBV) or hepatitis C virus (HCV).  HIV-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial. For patients with evidence of chronic HBV infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. Individuals with a history of HCV infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load Subjects with invasive malignancy over the previous year except treated early stage carcinomas of the skin, completely resected intraepithelial carcinoma of the cervix, completely resected papillary thyroid and follicular thyroid cancers, and localized prostate cancer treated with curative intent with surgery or radiation