Rivers provide critical water supply for many human societies and ecosystems, yet global knowledge of their flow rates is poor. We show that useful estimates of absolute river discharge (in cubic meters per second) may be derived solely from satellite images, with no ground-based or a priori information whatsoever. The approach works owing to discovery of a characteristic scaling law uniquely fundamental to natural rivers, here termed a river’s at-many-stations hydraulic geometry. A first demonstration using Landsat Thematic Mapper images over three rivers in the United States, Canada, and China yields absolute discharges agreeing to within 20–30% of traditional in situ gauging station measurements and good tracking of flow changes over time. Within such accuracies, the door appears open for quantifying river resources globally with repeat imaging, both retroactively and henceforth into the future, with strong implications for water resource management, food security, ecosystem studies, flood forecasting, and geopolitics.