We present TRICERATOPS, a new Bayesian tool that can be used to vet and validate TESS Objects of Interest (TOIs). We test the tool on 68 TOIs that have been previously confirmed as planets or rejected as astrophysical false positives. By looking in the false-positive probability (FPP)-nearby false-positive probability (NFPP) plane, we define criteria that TOIs must meet to be classified as validated planets (FPP < 0.015 and NFPP < 10-3), likely planets (FPP < 0.5 and NFPP < 10-3), and likely nearby false positives (NFPP > 10-1). We apply this procedure on 384 unclassified TOIs and statistically validate 12, classify 125 as likely planets, and classify 52 as likely nearby false positives. Of the 12 statistically validated planets, 9 are newly validated. TRICERATOPS is currently the only TESS vetting and validation tool that models transits from nearby contaminant stars in addition to the target star. We therefore encourage use of this tool to prioritize follow-up observations that confirm bona fide planets and identify false positives originating from nearby stars.