Pathogenic flaviviruses antagonize host cell Janus kinase/signal transducer and activator of transcription (JAK/STAT) signaling downstream of interferons α/β. Here, we show that flaviviruses inhibit JAK/STAT signaling induced by a wide range of cytokines beyond interferon, including interleukins. This broad inhibition was mapped to viral nonstructural protein 5 (NS5) binding to cellular heat shock protein 90 (HSP90), resulting in reduced Janus kinase–HSP90 interaction and thus destabilization of unchaperoned JAKs (and other kinase clients) of HSP90 during infection by Zika virus, West Nile virus, and Japanese encephalitis virus. Our studies implicate viral dysregulation of HSP90 and the JAK/STAT pathway as a critical determinant of cytokine signaling control during flavivirus infection.