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 effective barrier contraception during the entire study treatment period and through 4 months after the last dose of study drug, or agree to completely abstain from heterosexual intercourse
* Adequate cardiac function as assessed clinically by history and physical examination
* Signed informed consent

Exclusion Criteria:

* Active serious infection not controlled by oral or intravenous antibiotics
* Known active hepatitis B. Patients with chronic hepatitis B who are on appropriate viral suppressive therapy may be allowed after discussion with the principal investigator (PI)
* History of acute pancreatitis within 1 year of study or history of chronic pancreatitis
* History of alcohol abuse
* Uncontrolled hypertriglyceridemia
* Active secondary malignancy other than skin cancer (e.g., basal cell carcinoma or squamous cell carcinoma) that in the investigator's opinion will shorten survival to less than 1 year
* Active grade III-V cardiac failure as defined by the New York Heart Association criteria
* Uncontrolled, or active cardiovascular disease, specifically including, but not restricted to: myocardial infarction (MI), stroke, or revascularization within 3 months; unstable angina or transient ischemic attack; congestive heart failure prior to enrollment, or left ventricular ejection fraction (LVEF) less than lower limit of normal per local institutional standards prior to enrollment; diagnosed or suspected congenital long QT syndrome; clinically significant atrial or ventricular arrhythmias as determined by the treating physician; prolonged corrected QT (QTc) interval on pre-entry electrocardiogram (\> 470 msec) unless corrected after electrolyte replacement. or approved by cardiologist; Significant venous or arterial thromboembolism including deep venous thrombosis or pulmonary embolism. Patients with a history of treated prior superficial or catheter associated phlebitis will not be considered as significant embolism and after discussion with principal investigator (PI) will not be excluded from eligibility. Uncontrolled hypertension (diastolic blood pressure \> 90 mmHg; systolic \> 140mmHg). Patients with hypertension should be under treatment on study entry to effect blood pressure control
* Taking any medications or herbal supplements that are known to be strong inhibitors of CYP3A4 within at least 14 days or 5 half-lives before the first dose of ponatinib in patients with newly diagnosed only
* History or presence of clinically relevant central nervous system (CNS) pathology such as epilepsy, childhood or adult seizure, paresis, aphasia, stroke, severe brain injuries, dementia, Parkinson's disease, cerebellar disease, organic brain syndrome, or psychosis. Patients with active CNS leukemia will NOT be excluded
* Current autoimmune disease or history of autoimmune disease with potential CNS involvement
* Treatment with any investigational antileukemic agents or chemotherapy agents in the last 7 days before study entry, unless full recovery from side effects has occurred or patient has rapidly progressive disease judged to be life-threatening by the investigator
* Pregnant and lactating women will not be eligible; women of childbearing potential should have a negative pregnancy test prior to entering on the study and be willing to practice methods of contraception. Women do not have childbearing potential if they have had a hysterectomy or are postmenopausal without menses for 12 months. In addition, men enrolled on this study should understand the risks to any sexual partner of childbearing potential and should practice an effective method of birth control
* History of significant bleeding disorder unrelated to cancer, including: diagnosed congenital bleeding disorders (e.g., von Willebrand's disease); and diagnosed acquired bleeding disorder within one year (e.g., acquired anti-factor VIII antibodies)
* Patients with documented significant pleural or pericardial effusions unless they are thought to be secondary to their leukemia