Vesta, the second most massive asteroid, has long been perceived as anhydrous. Recent studies suggesting the presence of hydrated minerals and past subsurface water have challenged this long-standing perception. Yet, direct geologic indications of water activity on Vesta's surface were unexpected. Herein we show evidence that transient water flowed on the surface, in a debris-flow-like process, and left distinctive geomorphologic features. Based on detailed analysis of highest-resolution (~20 m/pixel) images obtained by the Dawn spacecraft, we identify a class of locally occurring, interconnected and curvilinear gully networks on the walls of young (< hundreds of Ma) impact craters, ending in lobate deposits near the crater floors. As curvilinear systems only occur within impact craters, we propose that they formed by a particulate-dominated flow of transient water that was released from buried ice-bearing deposits by impact-induced heating and melting. This interpretation is in accordance with the occurrence of pitted terrain on lobate deposits and crater floors. Pitted terrain is proposed to result from the degassing of volatiles. The proposed buried ice-bearing deposits are likely localized in extent and may be currently extant in Vesta's subsurface. Together with the discovery of water evaporation on Ceres and water activity on several small asteroids, our results support the new paradigm that water is widespread in the asteroid belt.