Amazon forests, which store ∼50% of tropical forest carbon and play a vital role in global water, energy, and carbon cycling, are predicted to experience both longer and more intense dry seasons by the end of the 21st century. However, the climate sensitivity of this ecosystem remains uncertain: several studies have predicted large-scale dieback of the Amazon, whereas several more recent studies predict that the biome will remain largely intact. Combining remote-sensing and ground-based observations with a size- and age-structured terrestrial ecosystem model, we explore the sensitivity and ecological resilience of these forests to changes in climate. We demonstrate that water stress operating at the scale of individual plants, combined with spatial variation in soil texture, explains observed patterns of variation in ecosystem biomass, composition, and dynamics across the region, and strongly influences the ecosystem's resilience to changes in dry season length. Specifically, our analysis suggests that in contrast to existing predictions of either stability or catastrophic biomass loss, the Amazon forest's response to a drying regional climate is likely to be an immediate, graded, heterogeneous transition from high-biomass moist forests to transitional dry forests and woody savannah-like states. Fire, logging, and other anthropogenic disturbances may, however, exacerbate these climate change-induced ecosystem transitions.