A self-consistent microscopic approach to calculate non-equilibrium pair correlations in strongly interacting driven binary mixtures is presented. The theory is derived from the many-body Smoluchowski equation for interacting Brownian particles by employing Kirkwood's superposition approximation as a closure relation. It is shown that the pair correlations can exhibit notable anisotropy and a strong tendency to laning in the driving direction. Furthermore, there are strong indications that pair correlations are characterized by a long-range decay along the drive. The theoretical results are in good quantitative agreement with the complementary Brownian dynamics computer simulations.