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.