Cold dark matter (CDM) simulations predict a central cusp in the mass distribution of galaxies. This prediction is in stark contrast with observations of dwarf galaxies that show a central plateau or "core" in their density distribution. The proposed solutions to this core-cusp problem can be classified into two types. One invokes feedback mechanisms produced by the baryonic component of the galaxies and the other assumes that the properties of the dark matter particle depart from the CDM hypothesis. Here we propose an alternative yet complementary explanation. We argue that cores are unavoidable in the self-gravitating systems of maximum entropy that result from non-extensive statistical mechanics. Their structure follows from the Tsallis entropy, which is attributed to systems with long-range interactions. Strikingly, the mass density profiles predicted by such thermodynamic equilibrium match the observed cores without any adjustment or tuning. Thus, the principle of maximum Tsallis entropy explains the presence of cores in dwarf galaxies.