It is shown that a non-magnetic metallic band in the presence of an antiferromagnetic background coupled only by the exchange interaction develops a superconducting instability similar to the one described by BCS theory plus additional terms that strongly renormalize the spin excitation spectra. A Bardeen-Pines-like hole-hole interaction Hamiltonian, mediated by magnetic excitations, is deduced from a microscopic model of a fermion band and a spin band that interact with each other only via the exchange interaction. The theory shows the appearance of an attractive interaction when the Fermi velocity in the non-magnetic band is larger than the magnon velocity in the magnetic band. The electron-magnon scattering is suppressed by the appearance of a spin gap simultaneously with the superconducting state. Although this model may well describe a general class of materials to be discovered, the possibility that this theory could describe superconducting cuprates is discussed.