Pevonedistat, Azacitidine, Fludarabine Phosphate, and Cytarabine in Treating Patients With Relapsed or Refractory Acute Myeloid Leukemia or Myelodysplastic Syndrome

Inclusion Criteria: * Patients must have had histologic verification of AML at the original diagnosis. Patients must have one of the following: * Recurrent disease in \>= 1st relapse with \>= 5% blasts in the bone marrow (M2/M3) marrow OR immunophenotypic evidence of disease with \>= 0.1% blasts detected by flow cytometry, OR evidence of recurrent cytogenetic or molecular abnormalities consistent with relapse, with or without extramedullary disease * Refractory AML is defined as \>= 5% blasts in the bone marrow (M2/M3) after \>= 2 induction attempts (i.e., 2 cycles of chemotherapy) * Patients with advanced MDS, including MDS that has progressed to AML, and have experienced relapse or are refractory after \>= 1 course of induction therapy, are eligible * Karnofsky \>= 50% for patients \> 16 years of age and Lansky \>= 50 for patients =\< 16 years of age * Patients must have fully recovered from the acute toxic effects of all prior anti-cancer therapy and must meet the following minimum duration from prior anti-cancer directed therapy prior to enrollment. If after the required timeframe, the numerical eligibility criteria are met, e.g., blood count criteria, the patient is considered to have recovered adequately * Cytotoxic chemotherapy or other anti-cancer agents known to be myelosuppressive * \>= 14 days must have elapsed after the completion of other cytotoxic therapy, with the exception of hydroxyurea, for patients not receiving standard maintenance therapy. Additionally, patients must have recovered from all acute toxic effects of prior therapy * NOTE: Cytoreduction with hydroxyurea must be discontinued \>= 24 hours prior to the start of protocol therapy * Anti-cancer agents not known to be myelosuppressive (e.g. not associated with reduced platelet or absolute neutrophil \[ANC\] counts): \>= 7 days after the last dose of agent * Antibodies: \>= 21 days must have elapsed from infusion of last dose of antibody, and toxicity related to prior antibody therapy must be recovered to grade =\< 1 * Hematopoietic growth factors: \>= 14 days after the last dose of a long-acting growth factor (e.g. pegfilgrastim) or 7 days for short-acting growth factor. For agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur. The duration of this interval must be discussed with the study chair and the study-assigned research coordinator * Interleukins, interferons and cytokines (other than hematopoietic growth factors): \>= 21 days after the completion of interleukins, interferon or cytokines (other than hematopoietic growth factors) * Stem cell Infusions (with or without traumatic brain injury \[TBI\]): * Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion including donor leukocyte infusion (DLI) or boost infusion: \>= 84 days after infusion and no evidence of graft versus host disease (GVHD) * Autologous stem cell infusion including boost infusion: \>= 42 days * Cellular Therapy: \>= 30 days after the completion of any type of cellular therapy (e.g. modified T cells, natural killer \[NK\] cells, dendritic cells, etc.) * Radiation therapy (XRT)/external beam irradiation including protons: \>= 14 days after local XRT; \>= 42 days after TBI, craniospinal XRT or if radiation to \>= 50% of the pelvis; \>= 42 days if other substantial brain metastases (BM) radiation * Radiopharmaceutical therapy (e.g., radiolabeled antibody, 131 iodine \[I\]-metaiodobenzylguanidine \[MIBG\]): \>= 42 days after systemically administered radiopharmaceutical therapy * Patients must not have received prior exposure to MLN4924 (pevonedistat) * Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>= 60 ml/min/1.73 m\^2 or a serum creatinine based on age/gender as follows: * 1 month to \< 6 months; 0.4 (male and female) * 6 months to \< 1 year; 0.5 (male and female) * 1 to \< 2 years; 0.6 (male and female) * 2 to \< 6 years; 0.8 (male and female) * 6 to \< 10 years; 1 (male and female) * 10 to \< 13 years; 1.2 (male and female) * 13 to \< 16 years; 1.5 (male) and 1.4 (female) * \>= 16 years; 1.7 (male) and 1.4 (female) * Bilirubin (sum of conjugated + unconjugated) =\< upper limit of normal (ULN) for age * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN. For the purpose of this study, the ULN for serum glutamate pyruvate transaminase (SGPT) is 45 U/L * Shortening fraction of \>= 27% by echocardiogram, or * Ejection fraction of \>= 50% by echocardiogram or radionuclide angiogram * No ventricular or supraventricular arrhythmia on electrocardiogram (EKG) * Prolonged rate corrected QT (QTc) interval \< 500 msec * Pulse oximetry \> 94% on room air if there is clinical indication for determination (e.g. dyspnea at rest) * International normalized ratio (INR) =\< 1.5 * Hemoglobin \> 8.0 g/dL (may receive red blood cell \[RBC\] transfusions) * All patients and/or their parents or legally authorized representatives must sign a written informed consent. Assent, when appropriate, will be obtained according to institutional guidelines Exclusion Criteria: * Pregnant or breast-feeding women will not be entered on this study due to risks of fetal and teratogenic adverse events as seen in animal/human studies, OR because there is yet no available information regarding human fetal or teratogenic toxicities. Pregnancy tests must be obtained in girls who are post-menarchal. Males or females of reproductive potential may not participate unless they have agreed to use 1 highly effective and 1 additional effective (barrier) method of contraception at the same time for the duration of study therapy and for 4 months after the completion of MLN4924 (pevonedistat) administration. True abstinence, when this is in line with the preferred and usual lifestyle of the subject, is acceptable. Periodic abstinence (e.g., calendar, ovulation, symptothermal, post-ovulation methods), withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception * Investigational drugs: Patients who are currently receiving another investigational drug are not eligible * Anti-cancer agents: Patients who are currently receiving other anti-cancer agents are not eligible (except hydroxyurea, which may be continued until 24 hours prior to start of protocol therapy) * Anti-GVHD agents post-transplant: Patients who are receiving cyclosporine, tacrolimus or other systemic agents to prevent graft-versus-host disease post bone marrow transplant are not eligible for this trial. Topical immunosuppressive agents (e.g. topical steroids) are allowed. Physiologic replacement of hydrocortisone is allowed * Patients who are taking drugs that are strong CYP3A4 inducers and cannot be switched to alternative drugs 14 days prior to enrollment are not eligible. Strong inducers of CYP34 are not permitted during the study * Patients with known hepatitis B surface antigen seropositive or known or suspected active hepatitis C infection are not eligible. NOTE: Patients who have isolated positive hepatitis B core antibody (i.e. in the setting of negative hepatitis B surface antigen and negative hepatitis B surface antibody) must have an undetectable hepatitis B viral load. Patients who have positive hepatitis C antibody may be included if they have an undetectable hepatitis C viral load * Patients with known hepatic cirrhosis or severe pre-existing hepatic impairment are not eligible * Patients with uncontrolled high blood pressure (i.e., \>= 99% for age) are not eligible * Patients with any of the following diagnoses: * Acute promyelocytic leukemia * Down syndrome * Juvenile myelomonocytic leukemia * Patients who have a documented active uncontrolled infection are not eligible * History of allergic reactions attributed to compounds of similar chemical or biologic composition as the study agent * Patients with human immunodeficiency virus (HIV) are not eligible unless they meet all of the following criteria: * CD4 count \> 350 cell/mm\^3 * Undetectable viral load * Maintained on modern therapeutic regimens utilizing non-CYP-interactive agents * No history of acquired immunodeficiency syndrome (AIDS)-defining opportunistic infections * Female patients who intend to donate eggs (ova) during the course of this study or 4 months after receiving their last dose of study drug(s) are not eligible * Male patients who intend to donate sperm during the course of this study or 4 months after receiving their last dose of study drug(s) are not eligible