- Previously received an investigational antineoplastic agent for NSCLC. - Previously received any prior tyrosine kinase inhibitor (TKI), including ALK-targeted TKIs. - Previously received more than 1 regimen of systemic anticancer therapy for locally advanced or metastatic disease. - Received chemotherapy or radiation within 14 days of first dose of study drug, except stereotactic radiosurgery (SRS) or stereotactic body radiation therapy (SBRT). - Received anti-neoplastic monoclonal antibodies within 30 days of the first dose of study drug. - Had major surgery within 30 days of the first dose of study drug, minor surgical procedures such as catheter placement or minimally invasive biopsies are allowed. - Have been diagnosed with another primary malignancy other than NSCLC, except for adequately treated non-melanoma skin cancer or cervical cancer in situ; definitively treated non-metastatic prostate cancer; or participants with another primary malignancy who are definitively relapse-free with at least 3 years elapsed since the diagnosis of the other primary malignancy. - Have symptomatic CNS metastases (parenchymal or leptomeningeal) at screening or asymptomatic disease requiring an increasing dose of corticosteroids to control symptoms within 7 days prior to randomization. - Have current spinal cord compression (symptomatic or asymptomatic and detected by radiographic imaging). Participants with leptomeningeal disease and without cord compression are allowed. - Be pregnant, planning a pregnancy, or breastfeeding. - Have significant, uncontrolled, or active cardiovascular disease, as defined by the study protocol. - Have uncontrolled hypertension. - Have a history or the presence at baseline of pulmonary interstitial disease, drug-related pneumonitis, or radiation pneumonitis. - Have an ongoing or active infection. - Have a known history of human immunodeficiency virus (HIV) infection. - Have a known or suspected hypersensitivity to brigatinib or its excipients and/or crizotinib or its excipients. - Have malabsorption syndrome or other gastrointestinal (GI) illness or condition. - Have any condition or illness that, in the opinion of the investigator, would compromise participant's safety or interfere with the evaluation of the study drug.