Motivated by recent experiments on the organic superconductor λ−(BETS)2FeCl4 we study the Jaccarino-Peter effect (JPE), the earliest example of magnetic-field-induced superconductivity, from the point of view of current-density-functional theory. It is found that both Meissner (diamagnetic) and Pauli (paramagnetic) pair breaking are suppressed by an exchange-correlation contribution to the vector potential, arising at the sites of the magnetic ions. This explains a number of otherwise puzzling experimental observations, and sheds new light on earlier theories of the JPE.