We identify discontinuous Galerkin methods for second-order elliptic problems having superconvergence properties similar to those of the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods. These methods use polynomials of degree k for both the potential as well as the flux. We show that the approximate flux converges with the optimal order of k+1, and that the local averages of the approximate potential superconverge to the averages of the potential, with order k+2. We also apply an element-by-element post-processing of the approximate solution to obtain a new approximation of the potential. The new approximate solution of the potential converges with order k+2. We provide numerical experiments that support our theoretical results.