Dasatinib in Treating Patients With Recurrent or Metastatic Malignant Salivary Gland Tumors

Inclusion Criteria: * Histologically or cytologically confirmed malignant salivary gland tumor (MSGT), including one of the following histologic subtypes: * Adenoid cystic carcinoma (ACC) with c v-kit Hardy-Zuckerman 4 feline sarcoma viral oncogene homolog (Kit) overexpression or non-ACC MSGT that is not amenable to potentially curable surgery or radiation * c-KIT overexpression in ACC patients is defined as cluster of differentiation (CD) 117 staining by immunohistochemistry (IHC) in 25% of tumor cells * No stipulation for c-KIT overexpression is not required for non-ACC MSGT patients * Patients must have radiographically measurable disease; radiographically measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan * Patients must have evidence of disease progression (objective growth of existing tumors) within 4 months of study entry * No known brain metastases, unless patient meets both of the following criteria: * Neurologic status stable for \>= 8 weeks after completion of definitive local therapy (surgery or radiotherapy) * No neurologic dysfunction that would confound study results * Life expectancy greater than 12 weeks * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 OR Karnofsky performance status (PS) \>= 60% * Leukocytes \>= 3,000/mcL * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelet \>= 100,000/mcL * Hemoglobin \>= 9 g/dL * Serum calcium =\< 12.0 mg/dL * Total serum bilirubin within normal institutional limits * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 2.5 x institutional upper limit of normal * Creatinine within normal institutional limits OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; all women of childbearing potential must have a negative pregnancy test prior to receiving dasatinib; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier; at least 4 weeks must have elapsed since any major surgery * Patients may not be receiving any other investigational agents * No prior treatment with any other targeted agents that inhibit vascular endothelial growth factor receptor (VEGFR), Breakpoint cluster region (BCR) ABL proto-oncogene 1, non-receptor tyrosine kinase (ABL), c-Src, c-KIT, platelet-derived growth factor (PDGF) beta receptor, or ephrin type-A receptor 2 (EPHA2) (e.g., imatinib mesylate) * History of allergic reactions attributed to compounds of similar chemical or biologic composition to dasatinib * Patients with corrected QT interval (QTc) prolongation (defined as a QTc interval equal to or greater than 500 msec), serious ventricular arrhythmia (ventricular fibrillation or ventricular tachycardia greater than or equal to 3 beats in a row) or other significant echocardiogram (ECG) abnormalities are excluded * Patients with any condition (e.g., gastrointestinal tract disease resulting in an inability to take oral medication, requirement for intravenous \[IV\] alimentation, or active peptic ulcer disease) that impairs their ability to swallow and retain dasatinib tablets are excluded * Patients with any of the following conditions are excluded: * Serious or non-healing wound, ulcer, or bone fracture * History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscesses within the past 28 days * Any history of cerebrovascular accident (CVA) or transient ischemic attack within the past 12 months * History of myocardial infraction, cardiac arrhythmia, stable/unstable angina, symptomatic congestive heart failure, or coronary/peripheral artery bypass graft or stenting within the past 6 months * History of pulmonary embolism within the past 12 months * Ejection fraction less than institutional normal by echocardiograph (only required for patients with a known history of congestive heart failure, low ejection fraction, or clinical symptoms/findings consistent with congestive heart failure) * Patients taking medications that are potent inducers or inhibitors of cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4) liver enzyme will be determined following a review of their case by the Principal Investigator; every effort should be made to switch patients taking such agents or substances to other medications * Patients with known brain metastases should be excluded; patients with brain metastases with stable neurologic status following local therapy (surgery or radiation) for at least 8 weeks from definitive therapy and without neurologic dysfunction that would confound the evaluation of neurologic and other adverse events are eligible for participation; patients cannot be receiving enzyme inducing anti-convulsants including carbamazepine, phenobarbital, and phenytoin * Patients with uncontrolled intercurrent illness including, but not limited to, ongoing or active infections or psychiatric illness/social situations that would limit compliance with study requirements are ineligible * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with dasatinib * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible * Patients who have an active pleural or pericardial effusion of any grade * Diagnosis of any second malignancy within the last 5 years; basal cell carcinoma, squamous cell skin cancer, stage I carcinoma fully treated, or in situ carcinoma that have been adequately treated with no evidence of recurrent disease for 12 months will be eligible