We propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a distributed optimal control problem governed by an elliptic linear convection diffusion PDE. We use degree k polynomials to approximate the state, adjoint state, their fluxes, and the optimal control, and we show the approximations converge with order k + 1 in the L2 norm. Finally, we use a simple element-by-element postprocessing scheme to obtain new superconvergent approximations of the state, dual state and the control. We show the postprocessed variables converge with order k + 2 in the L2 norm. We present 2D and 3D numerical experiments to illustrate our theoretical results.