An accurate background cross-section method for nuclear reactor cell homogenization is developed that is based on the very precise moment expansion approximation to the neutron escape probability that we recently suggested. The moment expansion approximation is so devised to properly generalize the equivalence relation that the background cross-section method attains a high accuracy to <1% error for each individual resonance, yet still preserves the very features of universality and simplicity of Bondarenko's method.