Collecting in situ observations from remote, high mountain rivers presents major challenges, yet real-time, high temporal resolution (e.g., daily) discharge data are critical for flood hazard mitigation and river management. In this study, we propose a method for estimating daily river discharge (RD) based on free, operational remote sensing precipitation data (Tropical Rainfall Measuring Mission (TRMM), since 2001). In this method, an exponential filter was implemented to produce a new precipitation time series from daily basin-averaged precipitation data to model the time lag of precipitation in supplying RD, and a linear-regression relationship was constructed between the filtered precipitation time series and observed discharge records. Because of different time lags in the wet season (rainfall-dominant) and dry season (snowfall-dominant), the precipitation data were processed in a segmented way (from June to October and from November to May). The method was evaluated at two hydrological gauging stations in the Upper Brahmaputra (UB) river basin, where Nash–Sutcliffe Efficiency (NSE) coefficients for Nuxia (>0.85) and Yangcun (>0.80) indicate good performance. By using the degree-day method to estimate the snowmelt and acquire the time series of new active precipitation (rainfall plus snowmelt) in the target basins, the discharge estimations were improved (NSE > 0.9 for Nuxia) compared to the original data. This makes the method applicable for most rivers on the Tibetan Plateau, which are fed mainly by precipitation (including snowfall) and are subject to limited human interference. The method also performs well for reanalysis precipitation data (Chinese Meteorological Forcing Dataset (CMFD), 1980–2000). The real-time or historical discharges can be derived from satellite precipitation data (or reanalysis data for earlier historical years) by using our method.