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