The fraction of the asteroid population that survived since the Solar System formation has experienced numerous collisional, dynamical and thermal events, which have shaped their structures and orbital properties. Small asteroids are often considered to be rubble-pile objects, aggregates held together only by self-gravity or small cohesive forces (1; 2). The artificial impact experiment of JAXA’s Hayabusa2 mission on the surface of asteroid Ryugu (3) created a surprisingly large crater (≈14 m). This unexpected result suggests that at least the near-surface of the asteroid is controlled to a large extent by its rather weak gravity rather than strength. Due to the inability to re-create these impact conditions in laboratory experiments, this observed regime of low-gravity, low-strength cratering remained largely unexplored so far. In addition, the very large times scales involved in the crater growth made it impossible to numerically simulate these impact processes up to now. Here we use a novel approach to model the entire cratering process resulting from impacts on small, weak asteroids, which uses shock physics code calculations directly. We found that small impacts can significantly deform weak asteroids, causing global resurfacing at the same time. We also show that even very low asteroid cohesions can drastically influence the outcome of an impact and that the collisional life-time of the overall asteroid shapes is significantly lower than the traditionally used life-time based on catastrophic disruption events. Consequently, we predict that NASA’s Double Asteroid Redirection Test (DART) impact on Dimorhpos (4; 5) will not lead to a cratering event, as originally anticipated (i.e., 6; 7). Rather, the impact is going to change the global morphology of the asteroid, if the surface cohesion is less than ≈ 10 Pa. Our results, together with the future observations by the ESA’s Hera mission (8) will provide constraints regarding the evolution of the shapes and structures of small asteroids by sub-catastrophic impacts.