Artificial lattices have served as a platform to study the physics of unconventional superconductivity. We study semiconductor artificial graphene-a honeycomb superlattice imposed on a semiconductor heterostructure-which hosts the Dirac physics of graphene but with a tunable periodic potential strength and lattice spacing, allowing control of the strength of the electron-electron interactions. We demonstrate a new mechanism for superconductivity due to repulsive interactions which requires a strong lattice potential and a minimum doping away from the Dirac points. The mechanism relies on the Berry phase of the emergent Dirac fermions, which causes oppositely moving electron pairs near the Dirac points to interfere destructively, reducing the Coulomb repulsion and thereby giving rise to an effective attraction. The attractive component of the interaction is enhanced by a novel antiscreening effect which, in turn, increases with doping; as a result, there is a minimum doping beyond which superconducting order generically ensues. The dominant superconducting state exhibits a spatially modulated gap with chiral p-wave symmetry. Microscopic calculations suggest that the possible critical temperatures are large relative to the low carrier densities, for a range of experimentally realistic parameters.