The Yarlung Zangbo River (YZR) is a sizeable highland river on the Tibetan Plateau, and its runoff process is crucial for understanding regional water resource features and related ecological patterns. However, the runoff characteristics of the YZR Basin (YZRB) remain unclear, especially how it would react to climate change. This study comprehensively analyzed the runoff characteristics of the entire YZRB based on a validated distributed hydrological model (SWAT) coupled with a glacier module (SWAT-glac), identified the runoff components, and explored the climate–discharge relationship, with a particular focus on the relationships between glacier runoff and changes in precipitation and air temperature. The results indicate that the SWAT-glac model, with localized glacier parameters, accurately simulates the runoff processes due to regional differences in meteorological conditions and uneven glacier distribution. Summer runoff dominates the basin, contributing 46.2% to 57.9% of the total, while spring runoff is notably higher in the downstream sections than in other areas. Runoff components vary significantly across river sections; precipitation is the primary contributor to basin-wide runoff (23.4–59.5%), while glacier runoff contribution can reach up to 54.8% in downstream areas. The study found that underlying surface conditions, particularly glacier coverage, significantly influence runoff responses to meteorological changes. The correlation between runoff and precipitation is stronger at stations where rainfall predominates, whereas runoff shows greater sensitivity to air temperature in glacier-covered areas. These findings enhance the understanding of runoff processes in the YZRB and offer valuable insights for the sustainable management of water resources in similar basins under climate change.