Viral protein homeostasis depends entirely on the machinery of the infected cell. Accordingly, viruses can illuminate the interplay between cellular proteostasis components and their distinct substrates. Here, we define how the Hsp70 chaperone network mediates the dengue virus life cycle. Cytosolic Hsp70 isoforms are required at distinct steps of the viral cycle, including entry, RNA replication, and virion biogenesis. Hsp70 function at each step is specified by nine distinct DNAJ cofactors. Of these, DnaJB11 relocalizes to virus-induced replication complexes to promote RNA synthesis, while DnaJB6 associates with capsid protein and facilitates virion biogenesis. Importantly, an allosteric Hsp70 inhibitor, JG40, potently blocks infection of different dengue serotypes in human primary blood cells without eliciting viral resistance or exerting toxicity to the host cells. JG40 also blocks replication of other medically-important flaviviruses including yellow fever, West Nile and Japanese encephalitis viruses. Thus, targeting host Hsp70 subnetworks provides a path for broad-spectrum antivirals.