Pevonedistat and Belinostat in Treating Patients With Relapsed or Refractory Acute Myeloid Leukemia or Myelodysplastic Syndrome

Inclusion Criteria: * Patients must have one of the following, histologically or cytologically confirmed: * AML (non- acute promyelocytic leukemia \[APL\] AML) * AML that is relapsed or refractory to at least one prior line of therapy * MDS, must meet all of the following at the time of enrollment: * Higher risk MDS (intermediate-2 or high risk by the original International Prognostic Scoring System \[IPSS\]), and * Relapsed, refractory, or intolerant to at least one prior line of therapy containing a hypomethylating agent (deoxyribonucleic acid \[DNA\] methyltransferase inhibitor) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Total bilirubin =\< upper limit of normal (ULN) for the laboratory except in patients with Gilbert's syndrome. Patients with Gilbert's syndrome may enroll if direct bilirubin =\< 1.5 x ULN for the laboratory of the direct bilirubin * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN * Creatinine clearance within normal limits for the laboratory OR estimated glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2 appropriate to race for patients with creatinine levels above institutional normal * Known human immunodeficiency virus (HIV) positive patients who meet the following criteria will be considered eligible: * CD4 count \> 350 cells/mm\^3 * Undetectable viral load * Maintained on modern therapeutic regimens utilizing non-CYP-interactive agents * No history of acquired immune deficiency syndrome (AIDS)-defining opportunistic infections * If evidence of chronic hepatitis B virus (HBV) infection, HBV viral load must be undetectable on suppressive therapy, if indicated * If history of hepatitis C virus (HCV) infection, patients must be treated and have an undetectable HCV viral load * The effects of belinostat and/or MLN4924 (pevonedistat) on the developing human fetus are unknown. For this reason and because histone deacetylase inhibitors and NEDD8-activating enzyme (NAE) inhibitory agents are known to be teratogenic, women of child-bearing potential and men must use 1 highly effective method and 1 additional (barrier) method of contraception at the same time, from the time of signing the informed consent through 4 months after the last dose of study drug (female and male condoms should not be used together). Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of MLN4924 (pevonedistat) and belinostat administration * Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible Exclusion Criteria: * Clinical picture indicative of leukostasis or evidence of disseminated intravascular coagulopathy * Patients with uncontrolled coagulopathy or bleeding disorder * Systemic antineoplastic therapy or radiotherapy for other malignant conditions within 14 days before the first dose of any study drug, except for hydroxyurea * Uncontrolled high blood pressure (i.e., systolic blood pressure \> 180 mm Hg, diastolic blood pressure \> 95 mm Hg) * 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) * 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) * Ongoing toxicities \>= grade 2 from prior therapy, except those related to hydroxyurea (which is permitted through the first 5 days of study treatment) * APL (M3) * Active central nervous system (CNS) leukemia * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MLN4924 (pevonedistat) or belinostat * Stem cell transplant within previous 3 months prior to initiation of study therapy * Major surgical procedures =\< 28 days before beginning study treatment or minor surgical procedures =\< 7 days before beginning study treatment. No waiting required after placement of a vascular access device * Uncontrolled intercurrent illness or infection * Circulating blast count \> 50,000 mm\^3 within 7 days preceding enrollment * Current candidacy for a potentially curative allogeneic stem cell transplant, unless declined * Left ventricular ejection fraction (LVEF) \< 50% as assessed by echocardiogram or radionuclide angiography * Prolongation of the heart-rate corrected QT (QTc) interval \>= 450 ms (i.e., grade 1 or higher) on electrocardiogram (ECG) prior to initiation of study treatment. * If baseline QTc on screening ECG is \>= 450 ms (i.e., grade 1 or higher): * Check potassium and magnesium serum levels, and * Correct any identified hypokalemia and/or hypomagnesemia and repeat ECG to confirm QTc interval * For patients with baseline heart rate \< 60 beats per minute (bpm) or \> 100 bpm, manual measurement of QT interval by cardiologist is required, with Fridericia correction applied to that manual measurement to determine the QTc for eligibility consideration * Note: For patients with a heart rate of 60-100 bpm, manual measurement of QT interval and use of the Fridericia formula to determine QTc is NOT required * Known cardiopulmonary disease defined as: * Unstable angina * Congestive heart failure (New York Heart Association \[NYHA\] class III or IV) * Myocardial infarction (MI) within 6 months prior to first dose (patients who had ischemic heart disease such as acute chest syndrome \[ACS\], MI, and/or revascularization greater than 6 months before screening and who are without cardiac symptoms may enroll) * Symptomatic cardiomyopathy * Clinically significant pulmonary hypertension requiring pharmacologic therapy * Clinically significant arrhythmia defined as any of the following: * History of polymorphic ventricular fibrillation or torsade de pointes * Permanent atrial fibrillation (a fib), defined as continuous a fib for \>= 6 months * Persistent a fib, defined as sustaining a fib lasting \> 7 days and/or requiring cardioversion in the 4 weeks before screening * Grade 3 a fib defined as symptomatic and incompletely controlled medically, or controlled with device (e.g., pace maker), or ablation in the past 6 months * Patients with paroxysmal a fib or \< grade 3 a fib for a period of at least 6 months are permitted to enroll provided that their rate is controlled on a stable regimen * Known congenital long QT syndrome * Second degree atrioventricular (AV) block type II or third degree AV block * Ventricular rate \< 50 bpm or \> 120 bpm * Treatment with clinically significant metabolic enzyme inducers within 14 days before the first dose of the study drug. * Note: Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * Ongoing or planned treatment with strong inhibitors of UGT1A1 * Any known UGT1A polymorphism, heterozygous or homozygous * History of prior therapy with belinostat or MLN4924 (pevonedistat) * Active gastrointestinal (GI) conditions that might predispose to drug intolerance or poor drug absorption * Known hepatic cirrhosis * Known moderate to severe chronic obstructive pulmonary disease, interstitial lung disease, and pulmonary fibrosis * No other prior malignancy is allowed except for the following: * In situ cervical cancer, * Adequately treated basal cell or squamous cell skin cancer, * Adequately treated stage I or II cancer from which the patient is currently in complete remission, and * Any other cancer from which the patient has been disease-free for at least 1 year * Medical, psychological, or social condition that, in the opinion of the investigator, may increase the patient's risk, interfere with the patient's participation in the study, or hinder evaluation of study results * Pregnant or nursing. Women of childbearing potential must have a negative serum pregnancy test performed within 7 days prior to the start of study therapy. * Note: Pregnant women are excluded from this study because MLN4924 (pevonedistat) is a NEDD8 inhibitor with the potential for teratogenic or abortifacient effects and because belinostat may cause teratogenicity and/or embryo-fetal lethality by virtue of targeting actively dividing cells. Because there is an unknown but potential risk for adverse events (AEs) in nursing infants secondary to treatment of the mother with MLN4924 (pevonedistat) or belinostat, breastfeeding should be discontinued if the mother is treated with MLN4924 (pevonedistat)/belinostat