Ruxolitinib Phosphate or Dasatinib With Chemotherapy in Treating Patients With Relapsed or Refractory Philadelphia Chromosome-Like Acute Lymphoblastic Leukemia

Inclusion Criteria:

* Patients with previously treated B-cell ALL (relapsed and/or refractory after prior therapy)
* Bone marrow involvement with \>= 5% lymphoblasts
* Documented genetic lesion(s) known to confer susceptibility to inhibition by either ruxolitinib or dasatinib or cytokine receptor-like factor 2 (CRLF2) positivity by flow cytometry (for the ruxolitinib cohort)
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2
* Total bilirubin \< 2.0 mg/dL
* Serum glutamate pyruvate transaminase (SGPT) or serum glutamic oxaloacetic transaminase (SGOT) \< 3 x upper limit of normal (ULN)
* Creatinine \< 2 mg/dL
* Females of childbearing potential must have a negative serum or urine beta human chorionic gonadotropin (beta-hCG) pregnancy test result within 14 days prior to the first dose of study drugs and must agree to use an effective contraception method during the study and for 30 days following the last dose of study drug; females of non-childbearing potential are those who are postmenopausal greater than 1 year or who have had a bilateral tubal ligation or hysterectomy; appropriate methods of birth control include the following: any 2 of the following methods used together: birth control implants, injections, or pills (except for progesterone only pills), intrauterine device (IUD), vasectomy, tubal ligation, barrier method (female or male condom with spermicide, cervical cap with spermicide, diaphragm with spermicide); male condom with spermicide and diaphragm; male condom with spermicide and cervical cap; unacceptable methods of birth control include using no birth control, withdrawal, rhythm method, vaginal sponge, any barrier method that does not use spermicide, progesterone only pills, and using male and female condoms at the same time
* Males who have partners of childbearing potential must agree to use an effective contraceptive method during the study and for 30 days following the last dose of study drug
* Patients or their legally authorized representative must provide written informed consent

Exclusion Criteria:

* Burkitt's leukemia or lymphoma, T-cell ALL or lymphoblastic lymphoma
* Patients having undergone prior allogeneic stem cell transplant within 3 months or having active graft versus host disease
* Patient is pregnant or breastfeeding
* Patients with uncontrolled active infections (fever \>= 38 degrees Celsius \[C\], septic shock)
* Isolated extramedullary relapse (i.e. testicular, central nervous system)
* Current or chronic hepatitis B or C infection, or known seropositivity for human immunodeficiency virus (HIV)
* Concurrent chemotherapy (except intrathecal chemotherapy)
* Major surgery within 4 weeks prior to first study dose
* Systemic chemotherapy/radiotherapy/investigational therapy within 14 days (with the exception of hydroxyurea and steroids) prior to starting therapy; for patients receiving ALL maintenance with 6-mercaptopurine, methotrexate, vincristine, and steroids - these agents should be discontinued at least 48 hours prior to start of study drugs; for patients on oral targeted therapies (such as imatinib, dasatinib, ponatinib), - these agents should be discontinued at least 48 hours prior to start of study drugs
* Patients must have recovered from acute non hematologic toxicity (to =\< grade 1) of all previous therapy prior to enrollment
* Known active central nervous system (CNS) leukemia, as defined by unequivocal morphologic evidence of lymphoblasts in the cerebrospinal fluid (CSF), use of CNS-directed local treatment for active disease within the prior 28 days, symptomatic CNS leukemia (i.e., cranial nerve palsies or other significant neurologic dysfunction) within 28 days; patients may have history of CNS leukemic involvement if definitively treated with prior therapy and no evidence of active disease (defined as \>= 2 consecutive spinal fluid assessments with no evidence of disease) at the time of registration; prophylactic intrathecal chemotherapy is not a criterion for exclusion
* Patients with active heart disease (New York Heart Association \[NYHA\] class 3-4 as assessed by history and physical examination, unstable angina/stroke/myocardial infarction within the last 6 months)
* Patients with a cardiac ejection fraction (as measured by either multi-gated acquisition \[MUGA\] scan or echocardiogram) \< 40%; (Note: patients who have had prior anthracycline exposure of \> 250 mg/m\^2 may be eligible after discussion with the principal investigator \[PI\])
* Second malignancy other than basal cell or squamous cell carcinoma of the skin or in situ carcinoma of the cervix or the breast, unless they are successfully treated with curative intent for more than 2 years before entering the study
* Malabsorption syndrome or other conditions that preclude enteral route of administration
* Patients requiring strong cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inhibitors
* Other severe acute or chronic medical or psychiatric condition or laboratory abnormality that in the opinion of the investigator may increase the risk associated with study participation or investigational product administration or may interfere with the interpretation of study results and/or would make the patient inappropriate for enrollment into this study