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

This randomized phase II trial studies how well giving erlotinib hydrochloride and cabozantinib-s-malate alone or in combination works as second or third line therapy in treating patient with stage IV non-small cell lung cancer. Erlotinib hydrochloride and cabozantinib-s-malate may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether giving erlotinib hydrochloride together with cabozantinib-s-malate is more effective than erlotinib hydrochloride or cabozantinib-s-malate alone in treating 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 \= 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