The statistics of extrasolar planetary systems indicate that the default mode of planet formation generates planets with orbital periods shorter than 100 days, and masses substantially exceeding that of the Earth. When viewed in this context, the Solar System is unusual. Here, we present simulations which show that a popular formation scenario for Jupiter and Saturn, in which Jupiter migrates inward from a > 5 AU to a ~ 1.5 AU before reversing direction, can explain the low overall mass of the Solar System's terrestrial planets, as well as the absence of planets with a < 0.4 AU. Jupiter's inward migration entrained s ~ 10-100 km planetesimals into low-order mean-motion resonances, shepherding and exciting their orbits. The resulting collisional cascade generated a planetesimal disk that, evolving under gas drag, would have driven any pre-existing short-period planets into the Sun. In this scenario, the Solar System's terrestrial planets formed from gas-starved mass-depleted debris that remained after the primary period of dynamical evolution.