Change in seasonal snowfall and glaciers ablation control year-to-year variations in streamflows of the Upper Indus Basin (UIB) and hence ultimately impacts the water availability in downstream areas of UIB. This situation calls for an urgent response to study the long-term variations in runoff components in response to climate change. The current study investigates the spatiotemporal variations in runoff and runoff components in response to climate change to the streamflows of the Gilgit River from 1981 to 2020 by using the University of British Columbia Watershed Model (UBC WM). Three statistical indices such as the Nash–Sutcliffe efficiency (NSE), the coefficient of determination (R2), and the correlation coefficient (CC) were used to evaluate the performance of UBC WM in simulating the streamflows against observed streamflows. According to statistical indices, the UBC WM performed fairly well during both calibration (1981–2000: R2 = 0.90, NSE = 0.87, and CC = 0.95) and validation periods (2001–2015: R2 = 0.86, NSE = 0.83, and CC = 0.92). Trend analysis revealed a significant increase in all runoff components with large interannual variations in their relative contributions to streamflows from 1981 to 2020. From 1981 to 2020, the average relative contribution of snowmelt, glacier melt, rainfall-runoff, and baseflow was estimated to be 25%, 46%, 5%, and 24%, respectively to the streamflows of the Gilgit River. Seasonal analysis showed that about 86% of total runoff was contributed to the Gilgit River during the summer season (April–September) while only 14% in the winter season (October–March). Further analysis of runoff at a spatial scale revealed that approximately 76% of the total runoff of Gilgit River is generated between elevations from 3680 to 5348 m while 19% of total runoff is generated at an elevation <3680 m and only 5% at an elevation >5348 m. Moreover, it was observed that groundwater contribution from soil lower zone (i.e., 76%) to streamflows was found greater than soil upper zone (i.e., 24%). The outcomes of this study will help the water resource managers and hydrologists to manage the water resources in downstream areas of the UIB for local consumption, industrial use, and agriculture.