The spread of mosquito-borne Zika virus (ZIKV), which causes neurological disorders and microcephaly, highlights the need for countermeasures against sudden viral epidemics. Here, we tested the concept that drugs targeting host proteostasis provide effective antivirals. We show that different cytosolic Hsp70 isoforms are recruited to ZIKV-induced compartments and are required for virus replication at pre- and post-entry steps. Drugs targeting Hsp70 significantly reduce replication of different ZIKV strains in human and mosquito cells, including human neural stem cells and a placental trophoblast cell line, at doses without appreciable toxicity to the host cell. By targeting several ZIKV functions, including entry, establishment of active replication complexes, and capsid assembly, Hsp70 inhibitors are refractory to the emergence of drug-resistant virus. Importantly, these drugs protected mouse models from ZIKV infection, reducing viremia, mortality, and disease symptoms. Hsp70 inhibitors are thus attractive candidates for ZIKV therapeutics with the added benefit of a broad spectrum of action.