Low-Intensity Chemotherapy, Ponatinib and Blinatumomab in Treating Patients With Philadelphia Chromosome-Positive and/or BCR-ABL Positive Acute Lymphoblastic Leukemia

Inclusion Criteria: * Patients \>= 18 years of age with previously untreated Ph-positive ALL (either t(9;22) and/or BCR-ABL positive) (includes patients initiated on first cycle of hyper-CVAD before cytogenetics known. These patients could have received one or two cycles of chemotherapy with or without other TKIs and still eligible. * If they achieved CR, they are assessable only for event-free and overall survival, or * If they failed to achieve CR, they are assessable for CR, event-free, and overall survival. Patients \>= 18 years of age with relapsed/refractory Ph-positive ALL or lymphoid accelerated or blast phase chronic myelogenous leukemia (CML) * Performance status =\< 2 (Eastern Cooperative Oncology Group \[ECOG\] scale) * Total serum bilirubin =\< 2 x upper limit of normal (ULN), unless due to Gilbert's syndrome * Alanine aminotransferase (ALT) =\< 3 x ULN * Aspartate aminotransferase (AST) =\< 3 x ULN * Serum lipase and amylase =\< 1.5 x ULN * Creatinine =\< 2.0 mg/dl * Female patients who: are postmenopausal for at least 1 year before the screening visit, OR are surgically sterile, OR if they are of childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent through 4 months after the last dose of study drug, or agree to completely abstain from heterosexual intercourse * Male patients, even if surgically sterilized (i.e., status post-vasectomy), who: agree to practice effec…