In this paper, the Meshless Local Petrov-Galerkin (MLPG) method is applied to compute convection-dominated flow problems. The results of the MLPG method are compared with the results of the finite volume method. The results show that the first-order upwind (FUD) scheme exhibits the false diffusion at a larger-Peclet number; the QUICK scheme and the MLPG method can obtain much closed solutions; but they have small overshoots produced at larger-Peclet number. The results also indicate that the MLPG method is a highly effective and accurate numerical method to deal with convection-dominated flow problems and can eliminate the effect of the false diffusion.