Feedbacks between climate, erosion and tectonics influence the rates of chemical weathering reactions(1,2), which can consume atmospheric CO2 and modulate global climate(3,4). However, quantitative predictions for the coupling of these feedbacks are limited because the specific mechanisms by which climate controls erosion are poorly understood. Here we show that climate-dependent chemical weathering controls the erodibility of bedrock-floored rivers across a rainfall gradient on the Big Island of Hawai'i. Field data demonstrate that the physical strength of bedrock in streambeds varies with the degree of chemical weathering, which increases systematically with local rainfall rate. We find that incorporating the quantified relationships between local rainfall and erodibility into a commonly used river incision model is necessary to predict the rates and patterns of downcutting of these rivers. In contrast to using only precipitation-dependent river discharge to explain the climatic control of bedrock river incision(5,6), the mechanism of chemical weathering can explain strong coupling between local climate and river incision.