Reversing HIV-1 latency promotes killing of infected cells and is essential for cure strategies; however, no single latency reversing agent (LRA) or LRA combination have been shown to reduce HIV-1 latent reservoir size in persons living with HIV-1 (PLWH). Here, we describe an approach to systematically identify LRA combinations to reactivate latent HIV-1 using genome-wide CRISPR screens. Screens on cells treated with suboptimal concentrations of an LRA can identify host genes whose knockout enhances viral gene expression. Therefore, inhibitors of these genes should synergize with the LRA. We tested this approach using AZD5582, an activator of the noncanonical nuclear factor κB (ncNF-κB) pathway, as an LRA and identified histone deacetylase 2 (HDAC2) and bromodomain-containing protein 2 (BRD2), part of the bromodomain and extra-terminal motif (BET) protein family targeted by BET inhibitors, as potential targets. Using CD4+ T cells from PLWH, we confirmed synergy between AZD5582 and several HDAC inhibitors and between AZD5582 and the BET inhibitor, JQ1. A reciprocal screen using suboptimal concentrations of an HDAC inhibitor as an LRA identified BRD2 and ncNF-κB regulators, especially BIRC2, as synergistic candidates for use in combination with HDAC inhibition. Moreover, we identified and validated additional synergistic drug candidates in latency cell line cells and primary lymphocytes isolated from PLWH. Specifically, the knockout of genes encoding CYLD or YPEL5 displayed synergy with existing LRAs in inducing HIV mRNAs. Our study provides insights into the roles of host factors in HIV-1 reactivation and validates a system for identifying drug combinations for HIV-1 latency reversal.