We present fully quantum-mechanical magnetotransport calculations for short-period lateral superlattices with one-dimensional electrostatic modulation. A non-perturbative treatment of both magnetic field and modulation potential proves to be necessary to reproduce novel quantum oscillations in the magnetoresistance found in recent experiments in the resistance component parallel to the modulation potential. In addition, we predict oscillations of opposite phase in the resistance component perpendicular to the modulation not yet observed experimentally. We show that the new oscillations originate from the magnetic miniband structure in the regime of overlapping minibands.