Epstein-Barr virus (EBV) persistently infects people worldwide. Delivery of ∼170-kb EBV genomes to nuclei and use of nuclear membrane-less replication compartments (RCs) for their lytic cycle amplification necessitate evasion of intrinsic antiviral responses. Proteomics analysis indicates that, upon B cell infection or lytic reactivation, EBV depletes the cohesin SMC5/6, which has major roles in chromosome maintenance and DNA damage repair. The major tegument protein BNRF1 targets SMC5/6 complexes by a ubiquitin proteasome pathway dependent on calpain proteolysis and Cullin-7. In the absence of BNRF1, SMC5/6 associates with R-loop structures, including at the viral lytic origin of replication, and interferes with RC formation and encapsidation. CRISPR analysis identifies RC restriction roles of SMC5/6 components involved in DNA entrapment and SUMOylation. Our study highlights SMC5/6 as an intrinsic immune sensor and restriction factor for a human herpesvirus RC and has implications for the pathogenesis of EBV-associated cancers.