Combination Chemotherapy in Treating Patients With Relapsed or Refractory Acute Lymphoblastic Leukemia, Lymphoblastic Lymphoma, Burkitt Lymphoma/Leukemia, or Double-Hit Lymphoma/Leukemia

Inclusion Criteria: * Relapsed/refractory acute lymphoblastic leukemia (ALL) or lymphoblastic lymphoma (LL): * Relapsed and/or refractory Philadelphia negative acute lymphoblastic leukemia or lymphoblastic lymphoma (LL) (Lead-in and Phase II) * Relapsed and/or refractory Philadelphia positive acute lymphoblastic leukemia, Burkitt leukemia/lymphoma or "double-hit" leukemia/lymphoma (phase II only) * At least 21 days elapsed from prior systemic chemotherapy (at least 14 days elapsed from prior systemic chemotherapy in the setting of rapidly progressive disease without significant residual extramedullary toxicity). Hydroxyurea and dexamethasone permitted up to approximately 24 hours prior to the start of therapy. Interruption of tyrosine kinase inhibitor (TKI) not required in Ph positive ALL subset * Age older than 15 years * Eastern Cooperative Oncology Group (ECOG) performance status =\< 3 (There may be certain patients with performance status \[PS\] 3 in the context of rapidly proliferative/refractory ALL who would benefit from this regimen. We don't want to exclude such patients who may derive benefit from this salvage regimen) * Serum bilirubin =\< 1.5 mg/dL * Serum glutamate pyruvate transaminase (SGPT) =\< 3 x upper limit normal (ULN), with exception for Gilbert's syndrome * Estimated creatinine clearance or GFR (glomerular filtration rate) \>= 50 mL/min * Signed informed consent Exclusion Criteria: * Active \>= grade 3 peripheral neuropathy * Active hepatic graft-versus-host disease * Known positivity for hepatitis B or C * Pregnancy * Breast feeding