An efficient implicit procedure for the Discontinuous Galerkin (DG) method is developed utilizing a pointwise relaxation algorithm. In the pointwise relaxation, those contributions from the degrees of freedom in own computational cell are accounted for in the implicit matrix inversion. The resulting scheme is shown to be stable with very large CFL numbers for both the Euler and the Navier-Stokes equations for typical test problems. In order to achieve a faster convergence, efforts are also made to reduce computing time of the present method by utilizing a p-multigrid scheme and also by solving a simplified matrix instead of a fully loaded dense matrix in the implicit matrix inversion. A superior performance of the present implicit DG method on the parallel computer using up to 128 PEs is shown in terms of readily achievable scalability and high parallel efficiency. The RANS simulation of turbulent flowfield over AGARD-B model is carried out to show the convergence property and numerical stability of the present implicit DG method for engineering applications.