A high-order Discontinuous Galerkin method with Lagrange Multipliers (DGLM) is presented for the solution of steady and unsteady advection-diffusion problems in the high Peclet number regime. Unlike HDG methods, it operates directly on the second-order form the advection-diffusion equation and does not require stabilization. Like the Discontinuous Enrichment Method (DEM), it chooses the basis functions among the free-space solutions of the homogeneous form of the governing differential equation, and relies on Lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries. For a homogeneous problem, the design of arbitrarily high-order elements based on the proposed method is supported by a detailed mathematical analysis. For a non-homogeneous one, the approximated solution is locally decomposed into its homogeneous and particular parts. The homogeneous part is captured by the DGLM elements designed for a homogenous problem. The particular part is obtained analytically after the source term is projected onto an appropriate polynomial space. An a posteriori error estimator for the proposed method is also derived to enable adaptive mesh refinement. All theoretical results are illustrated by high-order numerical simulations of steady and unsteady problems with steep gradients.