Uveal melanoma (UM) is the most prevalent primary intraocular malignancy in adults, which preferentially metastasizes to the liver in approximately half of all cases. Metastatic UM is notoriously resistant to therapy and is almost uniformly fatal. UM metastasis is most strongly associated with mutational inactivation of the BAP1 tumor suppressor gene. Given the role of BAP1 in epigenetic regulation as the ubiquitin hydrolase subunit of the polycomb repressive deubiquitinase (PR-DUB) complex, we conducted high-throughput drug screening using a well-characterized epigenetic compound library to identify new therapeutic vulnerabilities. We identified several promising new lead compounds, in particular the extra-terminal domain protein (BET) inhibitor mivebresib (ABBV-075). Mivebresib significantly improved survival rates in a metastatic uveal melanoma xenograft mouse model and entirely prevented detectable metastases to the bones, spinal cord, and brain. RNA sequencing revealed a notable overlap between the genes and pathways affected by HDAC and BET inhibition, including the reversal of gene signatures linked to high metastatic risk and upregulation of genes associated with a neuronal phenotype. Together, we found that UM cells are particularly vulnerable to class I HDAC and BET inhibition, and highlight the BET inhibitor mivebresib as a promising candidate for further clinical evaluation.