Current interferon alpha-based treatment of hepatitis C virus (HCV) infection fails to cure a sizeable fraction of patients treated. The cause of this treatment failure remains unknown. Here using mathematical modelling, we predict treatment failure to be a consequence of the emergent properties of the interferon-signalling network. HCV induces bistability in the network, creating a new steady state where it can persist. Cells that admit the new steady state alone are refractory to interferon. Using a model of viral kinetics, we show that when the fraction of cells refractory to interferon in a patient exceeds a critical value, treatment fails. Direct-acting antivirals that suppress HCV replication can eliminate the new steady state, restoring interferon sensitivity and improving treatment response. Our study thus presents a new conceptual basis of HCV persistence and treatment response, elucidates the origin of the synergy between interferon and direct-acting antivirals, and facilitates rational treatment optimization.