Warm ocean waters drive rapid ice‐shelf melting in the Amundsen Sea. The ocean heat transport toward the ice shelves is associated with the Amundsen Undercurrent, a near‐bottom current that flows eastward along the shelf break and transports warm waters onto the continental shelf via troughs. Here we use a regional ice‐ocean model to show that, on decadal time scales, the undercurrent's variability is baroclinic (depth‐dependent). Decadal ocean surface cooling in the tropical Pacific results in cyclonic wind anomalies over the Amundsen Sea. These wind anomalies drive a westward perturbation of the shelf‐break surface flow and an eastward anomaly (strengthening) of the undercurrent, leading to increased ice‐shelf melting. This contrasts with shorter time scales, for which surface current and undercurrent covary, a barotropic (depth‐independent) behavior previously assumed to apply at all time scales. This suggests that interior ocean processes mediate the decadal ice‐shelf response in the Amundsen Sea to climate forcing.