In this paper the Discontinuous Galerkin (DG) (or Lesaint-Raviart) method as applied to the analysis of viscoelastic flows is stabilized by adding monotonicity enforcement as proposed originally by van Leer. By using an implicit/explicit time discretization scheme, monotonicity is easily established and convergence at high values of the Deborah number is achieved using the Phan-Thien-Tanner model with a wide variety of material parameters for the stick-slip benchmark problem.