Gemcitabine Hydrochloride, Dasatinib, and Erlotinib Hydrochloride in Treating Patients With Pancreatic Cancer That Is Metastatic or Cannot Be Removed by Surgery

Inclusion Criteria: * Cytologically or histologically confirmed pancreatic adenocarcinoma (excluding islet cell or ampullary tumors) that is metastatic or unresectable * Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * Patients may have received prior chemotherapy for advanced disease as long as it did not include gemcitabine; if patients received prior adjuvant therapy including gemcitabine, patients must be \> 6 months from the last dose of gemcitabine; patients must have recovered from side effects of prior therapy to grade =\< 1 as measured by National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) version (v) 4.0 * Patients may have received prior radiation presuming \> 4 weeks since last dose and measurable disease outside the radiation field * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 * Anticipated life expectancy of greater than 3 months * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin \< 2.5 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic pyruvate transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal OR =\< 5 x institutional upper limit of normal when liver metastases are present * 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 child-bearing 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; 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 * Patients must be able to swallow pills 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 (with the exception of alopecia and neuropathy); no radiation is allowed on study * Patients who are receiving any other investigational agents * Major surgical procedure within 4 weeks of treatment * Patients with known brain metastases * History of allergic reactions attributed to compounds of similar chemical or biologic composition to dasatinib, erlotinib or gemcitabine * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study, breastfeeding should be discontinued if the mother is treated with erlotinib or dasatinib * Patients with immune deficiency are excluded * Patients on potent cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inducers and inhibitors * Malabsorption syndrome or other condition that would interfere with intestinal absorption * Other active malignancy (with the exception of locally treated non-melanoma skin cancers) * Human immunodeficiency virus (HIV) positive patients who are on combination antiretroviral therapy * Patients may not have any clinically significant cardiovascular disease including the following: * Myocardial infarction or ventricular tachyarrhythmia within 6 months * Prolonged corrected QT (QTc) \> 480 msec (Fridericia correction) * Known ejection faction less than institutional normal * Major conduction abnormality (unless a cardiac pacemaker is present)