The Moon experienced an intense period of impacts about 4 Gyr ago. This cataclysm is thought to have affected the entire inner Solar System and has been constrained by the radiometric dating of lunar samples: 40Ar-39Ar ages reflect the heating and degassing of target rocks by large basin-forming impacts on the Moon. Radiometric dating of meteorites from Vesta and the H-chondrite parent body also shows numerous 40Ar-39Ar ages between 3.4 and 4.1 Gyr ago, despite a different dynamical context, where impacts typically occur at velocities too low to reset geochronometers. Here we interpret the 40Ar-39Ar age record in meteorites to reflect unusually high impact velocities exceeding 10kms-1. Compared with typical impact velocities for main-belt asteroids of about 5kms-1, these collisions would produce 100-1,000 times more highly heated material by volume. We propose that the 40Ar-39Ar ages between 3.4 and 4.1 Gyr ago from Vesta, the H-chondrite parent body and the Moon record impacts from numerous main-belt asteroids that were driven onto high-velocity and highly eccentric orbits by the effects of the late migration of the giant planets. We suggest that the bombardment persisted for many hundreds of millions of years and affected most inner Solar System bodies.