Nivolumab and Dasatinib in Treating Patients With Relapsed or Refractory Philadelphia Chromosome Positive Acute Lymphoblastic Leukemia

Inclusion Criteria: * Patients must have a histologically confirmed diagnosis of Ph+ ALL * Detection of one of the following must be present: * t(9;22)(q34;q11) or 3-way variant by metaphase cytogenetics * Breakpoint cluster region (BCR)-Abelson (ABL) positive status by molecular analysis with qualitative polymerase chain reaction (PCR) or fluorescence in situ hybridization (FISH) * Patients must have primary refractory ALL based on failure to achieve a hematologic or molecular remission after induction therapy with dasatinib and steroids or dasatinib and chemotherapy, or have relapsed after treatment with a tyrosine kinase inhibitor with or without chemotherapy * Note: Prior course of dasatinib and steroid induction therapy should have included dasatinib 140mg PO daily on days 1-84 and prednisone 60mg/m\^2 (capped at 120mg, or equivalent steroid dose) on days 1-28; if patients were unable to tolerate full steroid dose during induction therapy they will still be eligible * Note: Patients with refractory or relapsed disease in the central nervous system will be eligible * Prior chemotherapy or tyrosine kinase inhibitor (TKI) treatment, aside from dasatinib, must be \>= 7 days before first investigational agent dose * Patients must exhibit an Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 * Patients must have adequate organ and bone marrow function prior to registration, as defined below: * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \< 2 x institutional upper limit of normal (IULN) * Total bilirubin \< 2.0 x IULN (unless Gilbert syndrome has been diagnosed); if leukemia infiltration of the liver is suspected to be causing liver function abnormalities the patient will still be eligible with principal investigator (PI) approval * Creatinine \< 2 x IULN * Creatinine clearance \> 40 mL/min (measured by Cockroft-Gault) * Females of child-bearing potential (FOCBP) must have a negative pregnancy test within 7 days of registration * Note: A FOCBP is any woman (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) who meets the following criteria: * Has not undergone a hysterectomy or bilateral oophorectomy * Has had menses at any time in the preceding 12 consecutive months (and therefore has not been naturally postmenopausal for \> 12 months) * Women must not be breastfeeding at the time of study registration * Women and men of reproductive potential should agree to use two effective means of birth control * For women, contraception should continue for 23 weeks after the last dose of nivolumab and 12 weeks after the last dose of dasatinib to allow complete clearance of drug and its principal metabolites from the body * For men, contraception should continue for 31 weeks after nivolumab and 12 weeks after dasatinib * Patients must have the ability to understand and the willingness to sign a written informed consent prior to registration on study Exclusion Criteria: * Patients may not be receiving any other investigational agents within 5 half-lives of the drug (if known); if the half-life is not known, investigational agents should not be taken within two weeks * Patients are not eligible if they have an intolerance to most recent prior TKI (other than dasatinib) at the lowest possible effective dose, defined as a grade \>= 3 toxicity considered at least possibly related to that TKI; patients are also excluded if they are intolerant or allergic to dasatinib and discontinued prior therapy due to a \>= grade 2 treatment related adverse event * Patients must not have a history of a grade 4 anaphylactic reaction to monoclonal antibody therapy or known hypersensitivity reactions to drugs formulated with polysorbate 90 * Patients must not have had any prior therapy with an anti-PD-1, anti-programmed cell death 1 ligand 1 (PD-L1), anti-programmed cell death 1 ligand 2 (PD-L2), anti-cluster of differentiation (CD)137 or anti-cytotoxic t-lymphocyte-associated protein 4 ligand (CTLA-4) antibody (or any antibody or drug specifically targeting T-cell costimulation or checkpoint pathways; for questions or uncertainties, please contact the PI or quality assurance manager (QAM) * Patients who have had allogeneic hematopoietic stem cell transplant (HSCT) are not eligible if they meet any of the following: * transplant is within 2 months from cycle 1, day 1 (C1D1) * Has clinically significant graft-versus-host disease requiring treatment * Has \>= grade 3 persistent non-hematological toxicity related to the transplant * Concomitant use of strong inhibitors of the cytochrome p450, family 3, subfamily a, polypeptide 4 (CYP3A4) isoenzyme is not permitted; must have wash-out period of 5 times the half-life of the compound before first dasatinib dose * Concomitant use of QT prolonging agents strongly associated with torsades de pointes is not permitted * Patients who have a known dasatinib-resistant ABL-kinase mutation such as T315I are not eligible; for confirmation, please contact PI * Patients who have any serious or uncontrolled medical disorder that would impair the ability of the subject to receive protocol therapy are not eligible; these include, but are not limited to: * Active infection that is not well controlled * Known pleural or pericardial effusion at baseline * Clinically significant gastrointestinal disease or digestive dysfunction compromising absorption of dasatinib * Pulmonary arterial hypertension * Uncontrolled or significant cardiovascular disease, including: * Myocardial infarction within 6 months of enrollment date * Uncontrolled angina or congestive heart failure within 3 months of enrollment date * Left ventricular ejection fraction (LVEF) \< 40% * Significant cardiac conduction abnormality, including: * History of clinically significant ventricular arrhythmia (such as ventricular tachycardia, ventricular fibrillation, or torsades de pointes) * History of second or third degree heart block (except for second degree type 1) * Corrected QT (QTc) interval \> 500 msec, unless a cardiac pacemaker is present * Prior malignancy active within the previous 3 years, except for locally curable cancers that have been apparently cured, such as basal or squamous cell skin cancers, superficial bladder cancer, or carcinoma in situ of the prostate, cervix or breast * Subjects with active, known or suspected autoimmune disease; (Note: Subjects with vitiligo, type I diabetes mellitus, hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are permitted to enroll) * Psychiatric illness/social situations that would limit compliance with study requirements * Any other illness or condition that the treating investigator feels would interfere with study compliance or would compromise the patient's safety or study endpoints * Female patients who are pregnant or nursing are not eligible * Patients are not eligible if they have a known positive test for hepatitis B virus surface antigen (HBV sAg) or hepatitis C virus ribonucleic acid (hepatitis C virus \[HCV\] antibody) indicating acute infection; Note: Patients with evidence of chronic hepatitis B infection will be allowed to enroll if on appropriate suppressive medications under the direction of a hepatologist and with PI approval * Patients who are known to be positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS) are not eligible * Patients must not have live vaccine therapies for prevention of infectious diseases within 28 days of first nivolumab dose * Patients who are unable to swallow oral medication are not eligible * Patients with active autoimmune disease or history of autoimmune disease that might recur, which may affect vital organ function or require immune suppressive treatment including chronic prolonged systemic corticosteroids (defined as corticosteroid use of duration one month or greater), should be excluded; these include but are not limited to patients with a history of: * Immune related neurologic disease * Multiple sclerosis * Autoimmune (demyelinating) neuropathy * Guillain-Barre syndrome * Myasthenia gravis * Systemic autoimmune disease such as systemic lupus erythematosus (SLE) * Connective tissue diseases * Scleroderma * Inflammatory bowel disease (IBD) * Crohn's * Ulcerative colitis * Patients with a history of toxic epidermal necrolysis (TEN) * Stevens-Johnson syndrome * Anti-phospholipid syndrome NOTE: Subjects with vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are permitted to enroll