Cabozantinib-S-Malate and Erlotinib Hydrochloride in Treating Patients With Previously Treated Metastatic Non-Small Cell Lung Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed non-small cell lung cancer harboring an EGFR mutation; NOTE: EGFR mutational status will be known and assays performed in Clinical Laboratory Improvement Amendments (CLIA) certified laboratories will be accepted * Patients should have tumor tissue available for retrieval; tissue blocks or unstained slides from the time of original diagnosis are acceptable if repeat biopsy is not indicated * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Patients must have received prior EGFR TKI therapy for metastatic disease and have documented evidence of radiologic disease progression while on EGFR TKI as treatment immediately prior to enrollment; (patients may have received prior chemotherapy, and retreatment with erlotinib is allowed) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 60%) * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 × upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 × institutional upper limit of normal * Lipase \< 2.0 x ULN and no radiologic or clinical evidence of pancreatitis * Creatinine =\< 1.5 × ULN OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Hemoglobin \>= 9 g/dL * Serum albumin \>= 2.8 g/dL * Urine protein/creatinine ratio (UPCR) =\< 1 * Serum phosphorus \>= lower limit of normal (LLN) * Calcium \>= LLN * Magnesium \>= LLN * Potassium \>= LLN * Women of childbearing potential must have a negative pregnancy test at screening; women of childbearing potential include women who have experienced menarche and who have not undergone successful surgical sterilization (hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or are not postmenopausal; postmenopause is defined as amenorrhea \>= 12 consecutive months; note: women who have been amenorrheic for 12 or more months are still considered to be of childbearing potential if the amenorrhea is possibly due to prior chemotherapy, antiestrogens, ovarian suppression or any other reversible reason * Women of child-bearing potential and men must agree to use adequate contraception prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of XL184 (cabozantinib) administration; sexually active subjects (men and women) must agree to use medically accepted barrier methods of contraception (e.g., male or female condom) during the course of the study and for 4 months after the last dose of study drug(s), even if oral contraceptives are also used; all subjects of reproductive potential must agree to use both a barrier method and a second method of birth control during the course of the study and for 4 months after the last dose of study drug(s) * Prior to the first patient registration, this study must have institutional review board (IRB) approval; a copy of the IRB approval for each site involved must be given to the Data Coordinating Center at City of Hope * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * The subject has received cytotoxic chemotherapy (including investigational cytotoxic chemotherapy) or biologic agents (e.g., cytokines or antibodies) within 3 weeks, or nitrosoureas/ mitomycin C within 6 weeks before the first dose of study treatment * Prior treatment with XL184 (cabozantinib) or other MET/hepatocyte growth factor (HGF) inhibitor * The subject has received radiation therapy: * To the thoracic cavity, abdomen, or pelvis within 2 weeks before the first dose of study treatment, or has ongoing complications, or is without complete recovery and healing from prior radiation therapy * To bone or brain metastasis within 14 days before the first dose of study treatment * To any other site(s) within 28 days before the first dose of study treatment * The subject has received prior treatment with a small molecule kinase inhibitor or a hormonal therapy (including investigational kinase inhibitors or hormones) within 14 days or five half-lives of the compound or active metabolites, whichever is longer, before the first dose of study treatment; prior erlotinib is required and does not require a 14-day wash out * The subject has received any other type of investigational agent within 28 days before the first dose of study treatment * The subject has not recovered to baseline or Common Terminology Criteria for Adverse Events (CTCAE) =\< grade 1 from toxicity due to all prior therapies except alopecia and other non-clinically significant adverse events (AEs) * The subject has a primary brain tumor * The subject has active brain metastases or epidural disease; subjects with brain metastases previously treated with whole brain radiation or radiosurgery or subjects with epidural disease previously treated with radiation or surgery who are asymptomatic and do not require steroid treatment for at least 2 weeks before starting study treatment are eligible; baseline brain imaging with contrast-enhanced CT or MRI scans for subjects with known brain metastases is required to confirm eligibility * The subject has prothrombin time (PT)/ International Normalized Ratio (INR) or partial thromboplastin time (PTT) test \>= 1.3 x the laboratory ULN within 14 days before the first dose of study treatment * The subject requires concomitant treatment, in therapeutic doses, with anticoagulants such as warfarin or warfarin-related agents, heparin, thrombin or Factor Xa inhibitors, or antiplatelet agents (e.g., clopidogrel); low dose aspirin (=\< 81 mg/day), low-dose warfarin (=\< 1 mg/day), and prophylactic low molecular weight heparin (LMWH) are permitted * Strong cytochrome P450 (CYP)3A4 inducers and inhibitors should be avoided; selection of alternate concomitant medications with no or minimal CYP3A4 enzyme inhibition potential is recommended; because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated list such as http://medicine.iupui.edu/clinpharm/ddis/table.aspx; medical reference texts such as the Physicians' Desk Reference may also provide this information; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * The subject has experienced any of the following: * Clinically-significant gastrointestinal bleeding within 6 months before the first dose of study treatment * Hemoptysis of \>= 0.5 teaspoon (2.5 mL) of red blood within 3 months before the first dose of study treatment * Any other signs indicative of pulmonary hemorrhage within 3 months before the first dose of study treatment * Patients with active bleeding or pathologic conditions that carry high risk of bleeding, such as known bleeding disorder or coagulopathy * The subject has radiographic evidence of cavitating pulmonary lesion(s) * The subject has tumor in contact with, invading or encasing any major blood vessels * The subject has evidence of tumor invading the gastrointestinal (GI) tract (esophagus, stomach, small or large bowel, rectum or anus), or any evidence of endotracheal or endobronchial tumor within 28 days before the first dose of cabozantinib * The subject has uncontrolled, significant intercurrent or recent illness including, but not limited to, the following conditions: * Cardiovascular disorders including: * Congestive heart failure (CHF): New York Heart Association (NYHA) class III (moderate) or class IV (severe) at the time of screening * Concurrent uncontrolled hypertension defined as sustained blood pressure (BP) \> 140 mm Hg systolic, or \> 90 mm Hg diastolic despite optimal antihypertensive treatment within 7 days of the first dose of study treatment * Any history of congenital long QT syndrome * Any of the following within 6 months before the first dose of study treatment: * Unstable angina pectoris * Clinically-significant cardiac arrhythmias * Stroke (including transient ischemic attack \[TIA\], or other ischemic event) * Myocardial infarction * Thromboembolic event requiring therapeutic anticoagulation (Note: subjects with a venous filter \[e.g. vena cava filter\] are not eligible for this study) * Gastrointestinal disorders particularly those associated with a high risk of perforation or fistula formation including: * Any of the following within 28 days before the first dose of study treatment * Intra-abdominal tumor/metastases invading GI mucosa * Active peptic ulcer disease * Inflammatory bowel disease (including ulcerative colitis and Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis * Malabsorption syndrome * Any of the following within 6 months before the first dose of study treatment: * Abdominal fistula * Gastrointestinal perforation * Bowel obstruction or gastric outlet obstruction * Intra-abdominal abscess; note: Complete resolution of an intra-abdominal abscess must be confirmed prior to initiating treatment with cabozantinib even if the abscess occurred more than 6 months before the first dose of study treatment * Other disorders associated with a high risk of fistula formation including percutaneous endoscopic gastrostomy (PEG) tube placement within 3 months before the first dose of study therapy * Other clinically significant disorders such as: * Active infection requiring systemic treatment within 28 days before the first dose of study treatment * Serious non-healing wound/ulcer/bone fracture within 28 days before the first dose of study treatment * History of organ transplant * Concurrent uncompensated hypothyroidism or thyroid dysfunction within 14 days before the first dose of study treatment * History of major surgery as follows: * Major surgery within 8 weeks of the first dose of cabozantinib, with complete wound healing; (patients with ongoing wound healing or other complications will be excluded) * Minor surgery within 4 weeks of the first dose of cabozantinib; Pleurx catheter placement within 7 days of the first dose of cabozantinib * The subject is unable to swallow tablets * The subject has a corrected QT interval calculated by the Fridericia formula (QTcF) \> 500 ms within 28 days before treatment; note: if initial QTcF is found to be \> 500 ms, two additional electrocardiograms (EKGs) separated by at least 3 minutes should be performed; if the average of these three consecutive results for QTcF is =\< 500 ms, the subject meets eligibility in this regard * The subject is unable or unwilling to abide by the study protocol or cooperate fully with the investigator or designee * The subject has had evidence within 2 years of the start of study treatment of another malignancy which required systemic treatment * History of allergic reactions attributed to compounds of similar chemical or biologic composition to XL184 (cabozantinib) or erlotinib * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with XL184 (cabozantinib); these potential risks may also apply to other agents used in this study * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated