Dasatinib in Treating Patients With Chronic Lymphocytic Leukemia

Inclusion Criteria: * Written informed consent and Health Insurance Portability and Accountability Act (HIPAA) authorization for release of personal information * Eastern Cooperative Oncology Group (ECOG) =\< 2 * In vitro dasatinib sensitivity (IC 50 =\< 50 nm per MTS assay) * Diagnosis of CLL and must meet one of the following: * Relapsed CLL in all patients who have failed at least one prior treatment, regardless of risk group * OR De novo (treatment-naïve) patients age \>= 65 who are not candidates for or do not want to pursue aggressive chemotherapy treatment * Have indications for treatment or evidence of progressive disease defined as follows (1996 NCI working group): * Massive or progressive splenomegaly * Massive lymph nodes (\>= 10 cm), nodal clusters (\>= 10 cm), or progressive lymphadenopathy * Symptomatic anemia and/or thrombocytopenia (Rai stages III or IV disease) * Autoimmune hemolytic anemia and/or thrombocytopenia that are poorly responsive to corticosteroid therapy * Progressive lymphocytosis with increase in lymphocyte count of \>= 50% over a 2-month period or an anticipated doubling time of \< 6 months * Repeated episodes of infection * Have not received CLL treatment within the past 2 weeks * Total bilirubin \< 2.0 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) =\< 2.5 x ULN * Alanine aminotransferase (ALT) =\< 2.5 x ULN * Serum creatinine \< 2.0 x ULN * International normalized ratio (INR) =\< 1.2 * Platelet (Plt) count \> 30,000 * Ability to take oral medication (dasatinib must be swallowed whole) * No clinically significant infections as determined by the investigator * Normal corrected QT (QTc) interval (\< 450 msec) * Serum potassium and magnesium are within normal limits * Women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy test (sensitivity =\< 25 IU HCG/L) within 72 hours prior to the start of study drug administration * Persons of reproductive potential must agree to use an adequate method of contraception throughout treatment and for at least 4 weeks after study drug is stopped prior to study enrollment; women of childbearing potential must be advised of the importance of avoiding pregnancy during trial participation and the potential risk factors for an unintentional pregnancy * After consent, discontinuation ("washout period") of any medications known to contribute significantly to the risk of QT prolongation or interfere with cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) mediated drug metabolism * Patient agrees to discontinue St. John's Wort while receiving dasatinib therapy (discontinue St. John's Wort at least 5 days before starting dasatinib) * Patient agrees that IV bisphosphonates will be withheld for the first eight weeks of dasatinib therapy due to risk of hypocalcemia * Patient agrees to discontinue anti-coagulants and anti-platelet drugs; note: a low dose aspirin regimen (81mg QD) is permitted with close monitoring of the subject's platelets for a level \>= 50,000/mm\^3 Exclusion Criteria: * Patients may not receive concurrent chemotherapy, radiotherapy, or immunotherapy * Pleural or pericardial effusion of any grade * Uncontrolled angina, \> New York Heart Association (NYHA) class III congestive heart failure or myocardial infarction (MI) within 6 months prior to study enrollment * Diagnosed congenital long QT syndrome * Any history of clinically significant ventricular arrhythmias (such as ventricular tachycardia, ventricular fibrillation, or Torsades de pointes) * Prolonged QTc interval on pre-entry electrocardiogram (\> 450 msec) * Subjects who are detained or imprisoned are not eligible * History of significant bleeding disorder unrelated to cancer, including: * Diagnosed congenital bleeding disorders (e.g., von Willebrand's disease) * Diagnosed acquired bleeding disorder within one year (e.g., acquired anti-factor VIII antibodies) * May not take concomitant medications that are generally accepted to have a risk of causing Torsades de Pointes including: (patients must discontinue drug 7 days prior to starting dasatinib) * Quinidine, procainamide, disopyramide * Amiodarone, sotalol, ibutilide, dofetilide * Erythromycin, clarithromycin * Chlorpromazine, haloperidol, mesoridazine, thioridazine, pimozide * Cisapride, bepridil, droperidol, methadone, arsenic, chloroquine, domperidone, halofantrine, levomethadyl, sparfloxacin, lidoflazine * Patients already taking other CYP inducers or inhibitors other than those listed above are eligible for the study only after principal investigator (PI) review; dose adjustments and/or monitoring of drug levels where applicable will be made as needed by the PI after consultation with pharmacy * Women who: * Are unwilling or unable to use an acceptable method to avoid pregnancy for the entire study period and for at least 4 weeks after cessation of study drug, or * Have a positive pregnancy test at baseline, or * Are pregnant or breastfeeding