We consider nonlinear elliptic systems of divergence type. We provide a new method for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. This method is applied to both homogeneous and inhomogeneous systems, in the latter case with inhomogeneity obeying the natural growth condition. Our methods extend previous partial regularity results, directly establishing the optimal Hölder exponent for the derivative of a weak solution on its regular set. We also indicate how the technique can be applied to further simplify the proof of partial regularity for quasilinear elliptic systems.