Like many galaxies of its size, the Milky Way is a disk with prominent spiral arms rooted in a central bar, although our knowledge of its structure and origin is incomplete. Traditional attempts to understand our Galaxy's morphology assume that it has been unperturbed by major external forces. Here we report simulations of the response of the Milky Way to the infall of the Sagittarius dwarf galaxy (Sgr), which results in the formation of spiral arms, influences the central bar and produces a flared outer disk. Two ring-like wrappings emerge towards the Galactic anti-Centre in our model that are reminiscent of the low-latitude arcs observed in the same area of the Milky Way. Previous models have focused on Sgr itself to reproduce the dwarf's orbital history and place associated constraints on the shape of the Milky Way gravitational potential, treating the Sgr impact event as a trivial influence on the Galactic disk. Our results show that the Milky Way's morphology is not purely secular in origin and that low-mass minor mergers predicted to be common throughout the Universe probably have a similarly important role in shaping galactic structure.