Pathogenic flaviviruses (such as West Nile virus and Zika virus (ZIKV)) inhibit host cell interferon-responsive Janus kinase/signal transducer and activator of transcription (JAK/STAT) signalling to antagonize antiviral activity and promote pathogenesis, with diverse mechanisms of this inhibition proposed. Surprisingly, we revealed that flaviviruses inhibit JAK/STAT signalling in response to many different cytokines not just interferons. Flaviviruses drive this broad inhibition via a mechanism that involves viral nonstructural protein 5 (NS5) interaction with host heat shock protein 90 (HSP90) within infected cells. HSP90-NS5 interaction was associated with reduced HSP90-JAK1 interaction and a loss of abundance of several host kinase clients of this chaperone, including JAK family proteins. Decreased JAK protein abundance led to refractory signal transmission, with a failure to phosphorylate STAT proteins in response to cytokines. The unforeseen importance of this general blockade for disease phenotypes was illustrated by ZIKV inhibition of JAK/STAT signalling in response to cytokines critical for foetal brain development. Neural stem cells infected with ZIKV displayed early deficits in cytokine-driven gliogenic and neurogenic differentiation. We thus identified ZIKV dysregulation of host HSP90 function as a potential mechanism for virus-induced microencephaly. Moreover, constitutive JAK/STAT signalling in glioblastoma cells was shown to be blocked by flaviviruses, providing insight into the mechanism responsible for the anti-cancer properties of these viruses. This novel HSP90 interaction axis constitutes a promising target both for antiviral development (aiming to disrupt viral NS5 interaction) and for anti-cancer therapeutic design (mimicking viral HSP90 perturbation).