In this study, we investigated the application and the transferability of the Soil Water and Assessment Tool (SWAT) in a partly glacierized Alpine catchment characterized by extreme climatic conditions and steep terrain. The model was initially calibrated for the 10 km2 watershed of the Damma glacier Critical Zone Observatory (CZO) in central Switzerland using monitoring data for the period of 2009–2011 and then was evaluated for 2012–2013 in the same area. Model performance was found to be satisfactory against both the Nash–Sutcliffe criterion (NS) and a benchmark efficiency (BE). The transferability of the model was assessed by using the parameters calibrated on the small watershed and applying the model to the approximately 100 km2 catchment that drains into the hydropower reservoir of the Göscheneralpsee and includes the Damma glacier CZO. Model results were compared to the reservoir inflow data from 1997 to 2010 and it was found that the model predicted successfully snowmelt timing and autumn recession but could not accurately capture the peak flow for certain years. Runoff was slightly overestimated from late May to June, when it is dominated by snowmelt. Finally, we investigated the response of the greater catchment to climate change using three different climate change scenarios, and the results were compared to those of a previous study, where two different hydrological models, PREVAH and ALPINE3D, were used. The methodology presented here, where SWAT is calibrated for a small watershed and then applied for a bigger area with similar climatic conditions and geographical characteristics, could work even under extreme conditions like ours. However, greater attention should be given to the differences between glacier melt and snowmelt dynamics. In conclusion, this assessment test on the transferability of SWAT on different scales gave valuable information about the strengths and weaknesses of the model when it was applied under conditions different to those under which it was calibrated.