A discontinuous Galerkin method with enrichment and Lagrange multipliers (DGLM) is proposed for the solution of problems with boundary layers. Specifically, this includes the steady and unsteady advection-diffusion equation with a spatially-varying advection field and the steady incompressible Navier-Stokes equations. The standard finite element method (FEM) is susceptible to the issue of spatial instability at practical mesh resolutions, typically observed as non-physical oscillations in the numerical solution to these problems. This is especially the case in the advection-dominated regime in which the boundary layers have steep gradients. Like the discontinuous enrichment method (DEM), the DGLM overcomes this issue through the use of novel shape functions designed to resolve boundary layers. These are chosen here element-wise as polynomials that are additively enriched with approximate free-space solutions of the governing differential equation. The enrichment functions are inspired by the boundary layer theory and are derived using an asymptotic analysis for different types of boundary layers. Inter-element solution continuity is weakly enforced using polynomial Lagrange multipliers. The method is shown to be stable in the inf-sup sense. Numerical results reveal that the DGLM has a lower error constant than the FEM and outperforms it for both the advection-diffusion equation in the high Peclet number regime and the incompressible Navier-Stokes equations in the laminar flow regime.