Vorinostat, Fludarabine Phosphate, Cyclophosphamide, and Rituximab in Treating Patients With Previously Untreated Chronic Lymphocytic Leukemia or Small Lymphocytic Lymphoma

Inclusion Criteria: * Patients must have a confirmed diagnosis of CLL/SLL * Patients with previously untreated cluster of differentiation (CD)20+ CLL/SLL must have either Rai stage III/IV disease or be Rai stage I/II with evidence of disease activity as defined by the National Cancer Institute (NCI) 1996 guidelines; patients with SLL must be Stage III or IV per Ann Arbor staging system * Patient must have consented to participate in the study and signed and dated an appropriate institutional review board (IRB)-approved consent form that conforms to federal and institutional guidelines * Patient must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1 or 2 * Patient must have an anticipated (untreated) survival of at least 3 months * Female patient of childbearing potential has a negative serum pregnancy test beta-human chorionic gonadotropin (hCG) within 2 weeks prior to receiving the first dose of vorinostat * Female patient is either post menopausal, free from menses for \>= 2 years, surgically sterilized or willing to use 2 adequate barrier methods of contraception to prevent pregnancy or agrees to abstain from heterosexual activity throughout the study, starting with Visit 1 * Male patients not sterilized must be willing to use adequate barrier methods of contraception to prevent pregnancy or agrees to abstain from heterosexual activity throughout the study, starting with Visit 1 * Absolute Neutrophil Count (ANC) \>= 1,500/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 9 g/dL * Prothrombin time or international normalized ratio (INR) =\< 1.5 upper limit of normal (ULN) unless receiving therapeutic anticoagulation * Partial thromboplastin time (PTT) =\< 1.2 times the ULN unless the patient is receiving therapeutic anticoagulation * Potassium level within normal limits * Magnesium level within normal limits * Serum creatinine =\< 1.5 x ULN OR if creatinine is \> 1.5 ULN the calculated creatinine clearance must be \>= 60 mL/min * Serum total bilirubin =\< 1.5 times ULN; patients with Gilbert's disease or similar syndrome involving slow conjugation of bilirubin are eligible with total bilirubin \> 1.5 times upper limit of normal; principal investigator (PI) review and approval required for anything above 2 times ULN * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 ULN * Alkaline phosphatase =\< 2.5 ULN * Patients with cytopenias due to disease or pseudohyperkalemia that do not meet these criteria, will be considered eligible with review and approval by the PI or Co-PI prior to study entry Exclusion Criteria: * Patients who have received cytotoxic chemotherapy, radiation therapy, immunotherapy, or cytokine treatment prior to study entry for CLL/SLL; patients who have received systemic steroids within 1 week of study entry are excluded, except patients on maintenance steroid therapy for a noncancerous disease * Patients with active hemolysis * Patients must not require sustained transfusion support of blood products * Patients who have undergone treatment with either stem cell or bone marrow transplant * Patients with active obstructive hydronephrosis * Patients with evidence of any significant systemic illness, active hepatitis B infection, active viral hepatitis infection or other active infection at the time of study entry * Patients with New York Heart Association class III or IV heart disease symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or other serious illness, such as acute or chronic graft versus host disease, that would preclude evaluation * Patients with congenital long QT syndrome and patients taking antiarrhythmic medicines or other medicinal products that lead to QT prolongation will only be eligible if their baseline corrected QT (QTc) prolongation is =\< 500 msec * Patients with known human immunodeficiency virus (HIV) infection * Patients who are pregnant or nursing * Patients with known brain or leptomeningeal involvement by malignancy * Patients who have, in the opinion of the investigator, other medical, social, or psychosocial factors that may negatively impact compliance or their safety by participation in this study * Patient is currently participating or has participated in a study with an investigational compound or device within 30 days of initial dosing with study drugs(s) * Patient had prior treatment with an histone deacetylases (HDAC) inhibitor (e.g., romidepsin \[Depsipeptide\], NSC-630176, MS 275, LAQ-824, belinostat \[PXD-101\], LBH589, MGCD0103, CRA024781, etc); patients who have received compounds with HDAC inhibitor-like activity, such as valproic acid, as anti-tumor therapy should not enroll in this study; patients who have received such compounds for other indications, e.g. valproic acid for epilepsy, may enroll after a 30-day washout period * Patient with a history of a prior malignancy with the exception of cervical intraepithelial neoplasia; non-melanoma skin cancer; adequately treated localized prostate carcinoma with prostate-specific antigen (PSA) =\< 1.0; or who has undergone potentially curative therapy with no evidence of that disease for five years, and/or who is deemed at low risk for recurrence by his/her treating physician