River networks are a crucial component of the earth system because they regulate carbon and nutrient exchange between continents, the atmosphere, and oceans. Quantifying the role of river networks at broad spatial scales must accommodate spatial heterogeneity, discharge variability, and upstream-downstream connectivity. Allometric scaling relationships of cumulative biogeochemical function with watershed size integrate these factors, providing an approach for understanding the role of fluvial networks in the earth system. Here we demonstrate that allometric scaling relationships of cumulative river network function are linear (power exponent ~ 1) when biogeochemical reactivity is high and river discharges are low, but become increasingly superlinear (power exponent > 1) as reactivity declines or discharge increases. Superlinear scaling indicates that biogeochemical function of entire river networks within a watershed is an emergent property that increases disproportionately with increasing watershed size. Expanding observation networks will increase precision in riverine fluxes of carbon and nutrients estimated by allometric scaling functions.