We investigate a space-time finite element method for solving a parabolic advection-diffusion problem. We use the Banach-Nečas-Babuška theorem to show the well-posedness of the continuous Petrov-Galerkin variational formulation for this problem. A fully discrete finite-element scheme is analyzed using the standard Galerkin method and unstructured meshes. An optimal error estimate is established in a discrete energy norm under a globally regularity condition. Some numerical results corroborate our theoretical results.