Entinostat And Imatinib Mesylate In Treating Patients With Relapsed or Refractory Philadelphia Chromosome-Positive Acute Lymphoblastic Leukemia

Inclusion Criteria: * Patients must have histologically confirmed BCR-ABL1 associated (Ph+) acute lymphoblastic leukemia (ALL) with primary refractory or relapsed disease; demonstration of BCR-ABL1 in leukemia cells by one or more of the following is required: t(9;22)(q34;q11.2) cytogenetics; FISH for BCR-ABL1 fusion; RT-PCR for BCR-ABL1 fusion * Prior treatment with tyrosine kinase inhibitors (including imatinib, nilotinib and/or dasatinib) is allowed, although patients must be off any tyrosine kinase inhibitor for a minimum of 72 hours prior to beginning protocol therapy * ECOG performance status of 0, 1 or 2 * Total WBC =\< 150,000 with no evidence for ongoing or impending leukostasis * Total bilirubin =\< 2.0 mg/dL unless elevated due to Gilbert's, hemolysis or leukemic infiltration * Aspartate transaminase (AST)/alanine transaminase (ALT) =\< 2.5 × upper limit of normal (ULN) unless due to leukemic infiltration * Serum creatinine =\< 2.0 mg/dL or creatinine clearance \> 50 ml/min * Left ventricular ejection fraction (LVEF) \>= 45% as measured by echocardiogram (ECHO) or MUGA * Patients who have undergone stem cell transplantation (SCT), autologous or allogeneic, are eligible provided that they are \> 4 weeks from stem cell infusion, have no active GVHD, and meet other eligibility criteria * Patients who fail primary induction therapy or relapse after achieving complete remission (CR) are eligible if they are \> 3 weeks off cytotoxic chemotherapy and \> 2 weeks off radiation therapy; patients must be off biologic therapies including hematopoietic growth factors \> 1 week; if using hydroxyurea (HU), steroids, or other non-cytotoxics for blast count control, patient must be off for \> 24 hrs before starting protocol therapy; patients must have recovered from all acute toxicities from any previous therapy * Female patients of childbearing age must have negative pregnancy test; women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients may not be receiving any other investigational agents * Active CNS leukemia; patients with known previous CNS leukemia may continue to receive intrathecal therapy with ara-C, methotrexate, and/or thiotepa plus steroids as prophylaxis against reactivation of previous CNS disease * Patients may not have received previous treatment with entinostat or other HDAC inhibitors * History of allergic reactions attributed to compounds of similar chemical or biologic composition to entinostat or other agents used in study * Uncontrolled intercurrent illness including, but not limited to, ongoing or active untreated infection, symptomatic congestive heart failure, unstable angina pectoris, unstable cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with entinostat * HIV-positive patients on combination antiretroviral therapy are ineligible