Ibrutinib, Fludarabine Phosphate, Cyclophosphamide, and Obinutuzumab in Treating Patients With Chronic Lymphocytic Leukemia

Inclusion Criteria: * Patients with a diagnosis of CLL/small lymphocytic lymphoma (SLL), with mutated (\> 2% deviation from germ line) IGHV gene, who meet criteria to initiate first-line treatment per International Workshop on CLL Working Group (IWCLL) 2008 guidelines * Patients must not have received prior CLL-directed therapy * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Absolute neutrophil count \> 500 uL * Platelet count \> 50,000/uL * Serum total bilirubin =\< 1.5 x upper limit of normal (ULN) or =\< 3 x ULN for patients with Gilbert's disease * Estimated creatinine clearance \>= 30 mL/min (calculated or measured by 24 hour urine collection) * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN * Women 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; women of non-childbearing potential are those who are postmenopausal greater than 1 year or who have had a bilateral tubal ligation or hysterectomy; men 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 * Free of prior malignancies for 3 years with exception of patients diagnosed with basal cell or squamous cell carcinoma of the skin, or carcinoma "in situ" of the cervix or breast, who are eligible even if they are currently treated or have been treated and/or diagnosed in the past 2 years prior to study enrollment; if patients have another malignancy that was treated within the last 2 years, such patients can be enrolled, after consultation with the principal investigator, if the likelihood of requiring systemic therapy for this other malignancy within 2 years is less than 10%, as determined by an expert in that particular malignancy at MD Anderson Cancer Center * Patients or their legally authorized representative must provide written informed consent * Prothrombin time (PT)/international normalization ratio (INR) \< 1.5 x ULN * Partial thromboplastin time (PTT) \< 1.5 x ULN * Activated partial thromboplastin time (aPTT) \< 1.5 x ULN Exclusion Criteria: * Major surgery, radiotherapy, chemotherapy, biologic therapy, immunotherapy, experimental therapy within 3 weeks prior to the first dose of the study drugs and/or monoclonal antibody =\< 6 weeks prior to first administration of study treatment * Patients with del(17p) by FISH (or known tumor protein p53 \[TP53\] mutation) * Patients with unmutated (=\< 2% homology with germ line) IGHV * Uncontrolled active systemic infection (viral, bacterial, and fungal) * Significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, QT prolongation or familial history of QT prolongation, congestive heart failure, or myocardial infarction within 6 months of screening, or any class 3 or 4 cardiac disease as defined by the New York Heart Association Functional Classification * History of stroke or cerebral hemorrhage within 6 months * Patient is pregnant or breast-feeding * Current or chronic hepatitis B or C infection, or known seropositivity for human immunodeficiency virus (HIV); subjects who are positive for hepatitis B or C core antibody or hepatitis B or C surface antigen must have a negative polymerase chain reaction (PCR) result before enrollment; those who are PCR positive will be excluded; Note: Patients who are receiving intravenous immunoglobulins may become seropositive for hepatitis B antibodies; these patients are allowed on the study without additional testing * Active, uncontrolled autoimmune phenomenon (autoimmune hemolytic anemia or immune thrombocytopenia) requiring steroid therapy * Concurrent use of investigational therapeutic agent * Patients may not receive other concurrent chemotherapy, radiotherapy, or immunotherapy; localized radiotherapy to an area not compromising bone marrow function does not apply * Malabsorption syndrome or other condition that precludes enteral route of administration * Concomitant use of warfarin or other vitamin K antagonists * Requires treatment with a strong cytochrome P450 (CYP), family 3, subfamily A (3A) inhibitor * 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 * Known bleeding disorders (e.g., von Willebrand's disease) or hemophilia