Flaviviruses, including dengue virus (DENV) and Zika virus (ZIKV), cause severe human disease. Co-opting cellular factors for viral translation and viral genome replication at the endoplasmic reticulum is a shared replication strategy, despite different clinical outcomes. Although the protein products of these viruses have been studied in depth, how the RNA genomes operate inside human cells is poorly understood. Using comprehensive identification of RNA-binding proteins by mass spectrometry (ChIRP-MS), we took an RNA-centric viewpoint of flaviviral infection and identified several hundred proteins associated with both DENV and ZIKV genomic RNA in human cells. Genome-scale knockout screens assigned putative functional relevance to the RNA-protein interactions observed by ChIRP-MS. The endoplasmic-reticulum-localized RNA-binding proteins vigilin and ribosome-binding protein 1 directly bound viral RNA and each acted at distinct stages in the life cycle of flaviviruses. Thus, this versatile strategy can elucidate features of human biology that control the pathogenesis of clinically relevant viruses.