We consider a discontinuous Galerkin method to solve boundary value problems in curved boundary domains in two-dimensional. The question that arises concerns the reduction of the order of convergence of numerical methods when considering the approximation of the domain by a polygonal mesh. Unless the boundary conditions can be accurately transferred from the physical boundary to the computational boundary, the isoparametric element method is usually employed to recover the optimal convergence orders. However, this technique involves more complex algebra and additional computational costs when compared to approaches using polygonal meshes, which are widely used due to their simplicity in many applications. In this paper, we present and analyse a higher-order strategy that achieves the optimal convergence order on polygonal approximations of domains with smooth boundaries. The boundary approximation error is corrected by means of polynomial reconstructions of the boundary conditions. We present a study on the existence and uniqueness of the solution and derive error estimates for a two-dimensional linear reaction-diffusion boundary-value problem with homogeneous Dirichlet boundary conditions in convex and non-convex domains. We prove that the numerical solution exhibits an optimal convergence rate under certain regularity conditions on the solution. A numerical benchmark is provided to illustrate the theoretical results proven in this work.