Under the current global warming trend, droughts are expected to increase, with serious implications for water resources management. This study analyzed the regional aspects of droughts in terms of streamflow deficiencies over the Andean rivers of Patagonia, Argentina. Based on the variable threshold level method, the main characteristics of streamflow droughts were obtained for the hydrological years 1962/63–2014/15, considering three different severity levels over 11 representative basins. Two distinct regional behaviors were identified in terms of temporal variations of streamflow drought duration and its cumulative deficit volume, dividing the study area into North and Central Patagonia. The effects of the Southern Annular Mode (SAM), the El Niño–Southern Oscillation (ENSO), and the Pacific Decadal Oscillation (PDO) on the interannual and interdecadal variability of streamflow droughts were assessed through an empirical decomposition applied to the regional time series. These large-scale climatic oscillations have a distinct regional and temporal behavior in terms of the modulation of streamflow drought variability. Considering the interannual streamflow drought variability, the El Niño signal is more consistent and contributes with humid conditions, especially over North Patagonia. The multi-decadal component of the streamflow drought time series is linked to the upward trend in SAM, particularly over Central Patagonia.