Increasing demands for high-density, stable nanoscale memory elements, as well as fundamental discoveries in the field of spintronics, have led to renewed interest in exploring the coupling between magnetism and electric fields. Although conventional magnetoelectric routes often result in weak responses, there is considerable current research activity focused on identifying new mechanisms for magnetoelectric coupling. Here we demonstrate a linear magnetoelectric effect that arises from a carrier-mediated mechanism, and is a universal feature of the interface between a dielectric and a spin-polarized metal. Using first-principles density functional calculations, we illustrate this effect at the SrRuO3/SrTiO3 interface and describe its origin. To formally quantify the magnetic response of such an interface to an applied electric field, we introduce and define the concept of spin capacitance. In addition to its magnetoelectric and spin capacitive behaviour, the interface displays a spatial coexistence of magnetism and dielectric polarization, suggesting a route to a new type of interfacial multiferroic.