Erlotinib Hydrochloride and Cabozantinib-s-Malate Alone or in Combination as Second or Third Line Therapy in Treating Patients With Stage IV Non-small Cell Lung Cancer

Criteria: * Tumor with a sensitizing mutation in epidermal growth factor receptor (EGFR), defined as follows: * EGFR mutation testing of tumor has been performed and did not demonstrate an EGFR tyrosine kinase inhibitor sensitizing mutation; at minimum, testing for EGFR exon 19 deletion and exon 21 L858R mutations must have been included; OR * EGFR mutation testing has been attempted and is inconclusive (for example, due to lack of sufficient deoxyribonucleic acid \[DNA\] yield); OR * EGFR mutation status is unknown but tumor is positive for at least one alternative driver mutation, i.e: Kirsten rat sarcoma viral oncogene homolog (KRAS) mutation, v-raf murine sarcoma viral oncogene homolog B (BRAF) mutation, human epidermal growth factor receptor 2 (HER2) mutation, ret proto-oncogene (RET) rearrangement/fusion, or one not listed following approval by the study chair prior to registration Inclusion Criteria: * INCLUSION CRITERIA STEP 1: * Cytologically or histologically confirmed non-small cell lung carcinoma (NSCLC) * Predominant non-squamous histology (patients with NSCLC not otherwise specified \[NOS\] are eligible); mixed tumors will be categorized by the predominant cell type; if small cell elements are present the patient is ineligible * Stage IV disease (includes M1a, M1b, or recurrent disease), according to the 7th edition of the lung cancer TNM classification system * Patients must have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) version (v) 1.1 criteria; baseline measurements and evaluation of all sites of disease must be obtained within 4 weeks prior to registration * Prior to registration, the investigator/site must confirm that sufficient pathology material representative of patient's cancer is available for submission for MET immunohistochemical (IHC) testing * Patients must have received one or two lines of prior chemotherapy (first line platinum-doublet based chemotherapy plus switch maintenance chemotherapy counts as one line of therapy); prior adjuvant chemotherapy for early stage disease does not count as one line of therapy if 12 months or greater elapsed between completion of adjuvant therapy and initiation of first-line systemic therapy; if less than 12 months elapsed, adjuvant chemotherapy counts as one line of therapy * Any prior chemotherapy (based on administration schedule) must have been completed in greater than or equal to the time frames specified in the protocol * Patients must have discontinued treatment with any other type of investigational agent \>= 4 weeks prior to registration * Patients must have recovered to baseline or Common Terminology Criteria for Adverse Events (CTCAE) v 4.0 =\< grade 1 from toxicity due to all prior therapies except alopecia and other non-clinically significant adverse events (AEs) * Patients with no known brain metastasis at baseline must have baseline brain imaging within 12 weeks prior to study registration not demonstrating brain metastases; patients with brain metastases at baseline must have baseline brain imagining within 4 weeks prior to study registration and meet all of the specific criteria for brain mets listed in the protocol * Radiation related toxicities must have resolved to =\< grade 1 prior to registration * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status between 0-2 * Patients must have an anticipated life expectancy greater than 3 months * Acceptable bone marrow, renal and hepatic function within 2 weeks prior to registration as defined in the protocol * Patients must have corrected QT interval calculated by the Fridericia formula (QTcF) =\< 500 ms within 28 days before registration * Patients must be able to swallow tablets * INCLUSION CRITERIA STEP 2: * Patients must have met all eligibility requirements for Step 1 at time of registration to Step 1 to be eligible for Step 2 * Patients must have radiographic progressive disease per RECIST v1.1 criteria after \>= 2 courses of therapy on Arm A or Arm B * Patients must be registered to Step 2 within 4 weeks of the last dose of treatment administration from Step 1 * Patients must have an ECOG performance status between 0-2 * Patients must have recovered to baseline (pre-Step 1) or CTCAE version 4.0 \<= grade 1 from toxicity due to all prior therapies except alopecia and other non-clinically significant AEs Exclusion Criteria: * EXCLUSION CRITERIA STEP 1: * Patients without sufficient pathology material representative of the patient's cancer (tumor block or 10 unstained slides) * Prior erlotinib, other EGFR tyrosine kinase inhibitor therapy, vascular endothelial growth factor receptor (VEGFR) tyrosine kinase inhibitor therapy, Met tyrosine kinase inhibitor therapy, or Met monoclonal antibody (MetMAb); prior antibody therapy such as bevacizumab or cetuximab is allowed with a washout period depending on dosing interval and investigational nature * Prior radiation therapy to the thoracic cavity, abdomen, or pelvis within 3 months prior to registration, to bone or brain metastasis within 14 days prior to registration, or to any other site within 28 days prior to registration * History of the following: Clinically-significant gastrointestinal (GI) bleeding within 6 months prior to registration; Hemoptysis of \>= 0.5 teaspoon (2.5 mL) of red blood within 3 months prior to registration; Any other signs indicative of pulmonary hemorrhage within 3 months prior to registration * Radiographic or other evidence of the following within 28 days prior to registration: • Tumor invading the GI tract (esophagus, stomach, small or large bowel, rectum or anus), or any evidence of endotracheal or endobronchial tumor; Cavitating pulmonary lesion(s); Tumor in contact with, invading or encasing any major blood vessels * Psychiatric illness/social situations that would limit compliance with study requirements * History of major thrombotic events (deep vein thrombosis \[DVT\] or pulmonary embolism \[PE\]) within 6 months prior to registration * Concomitant treatment, in therapeutic doses, with anticoagulants such as warfarin or warfarin-related agents, heparin, low molecular weight heparin (LMWH), thrombin or Factor Xa inhibitors, or antiplatelet agents (e.g., clopidogrel). (low dose aspirin \[=\< 81 mg/day\] and prophylactic LMWH are permitted) * Concomitant treatment of strong cytochrome P450 3A4 (CYP3A4) inducers (e.g., dexamethasone, phenytoin, carbamazepine, rifampin, rifabutin, rifapentine, phenobarbital, and St. John's wort) * 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; Any history of congenital long QT syndrome; Any of the following within 6 months prior to registration: * Unstable angina pectoris * Clinically-significant cardiac arrhythmias * Stroke (including transient ischemic attack \[TIA\], or other ischemic event) * Myocardial infarction * GI disorders particularly those associated with a high risk of perforation or fistula formation specified in the protocol * Other disorders associated with a high risk of fistula formation including percutaneous endoscopic gastrostomy (PEG) tube placement within 3 months prior to registration * Uncontrolled, significant, intercurrent or recent illness * Prior malignancy within 2 years prior to registration which required systemic treatment or is currently active * Pregnant or breast-feeding * Patients with known human immunodeficiency virus (HIV) disease taking antiretroviral therapy * Known chronic active hepatitis B * EXCLUSION CRITERIA (STEP 2): * Intervening anticancer treatment or major surgical procedure(s) between Step 1 and Step 2, except palliative radiation to the bone finishing \>= 2 weeks prior to registration to Step 2 * Central nervous system (CNS) progression; patients with stable CNS disease are allowed * Intercurrent illness or disease complication that the investigator believes would limit the ability to safely tolerate the combination of erlotinib and cabozantinib