Venetoclax, Dasatinib, Prednisone, Rituximab and Blinatumomab for the Treatment of Newly Diagnosed or Relapsed Philadelphia Chromosome Positive Acute Lymphoblastic Leukemia or Mixed Phenotype Acute Leukemia

Inclusion Criteria:

* 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. Participants with MPAL will be enrolled for induction therapy only
* 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 CYP3A inhibitors (such as fluconazole, ketoconazole and clarithromycin) prior to enrollment
* At least 3 half-lives must have passed before cycle 1 day 3 (C1D3) for participants that have used moderate CYP3A inhibitors (such as fluconazole, ketoconazole and clarithromycin), strong or moderate CYP3A inducers (such as rifampin, carbamazepine, phenytoin, and St. John's wort), P-glycoprotein (P-gp) inhibitors, or warfarin prior to enrollment
* At least 4 weeks must have passed before (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

Exclusion Criteria:

* For newly diagnosed subjects: who have received treatment with cytotoxic chemotherapy, radiotherapy or immunotherapy for their ALL/MPAL, or prior dasatinib treatment. For relapsed subjects: prior dasatinib treatment (however treatment with other tyrosine kinase inhibitors \[TKIs\] is permitted)
* 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 or three half-lives of enrollment (i.e. initiation of dasatinib)
* Subjects with chronic myelogenous leukemia (CML) in myeloid blast crisis, Ph+ acute myeloid leukemia (AML) or acute leukemia lineage that cannot be classified based on existing criteria (e.g., from the World Health Organization \[WHO\] or International Consensus Classification \[ICC\])
* Subjects with clinically serious infections as determined by the provider requiring ongoing antibiotic therapy. This does not include antibiotic treatment for neutropenic fever
* Individuals 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 stem cell transplant must be at least 100 days after transplant, and without active treatment for graft versus host disease (GVHD) other than topical medications
* 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 persons 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 individuals on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial.
  * For individuals 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 individuals 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
* Subjects with any gastrointestinal condition which would lead to inability to absorb an oral medication