This paper presents an accuracy-preserving p-weighted limiter for discontinuous Galerkin methods on one-dimensional and two-dimensional triangular grids. The p-weighted limiter is the extension of the second-order WENO limiter by Li et al. [W. Li, J. Pan and Y-X Ren, Journal of Computational Physics 364(2018)314-346] to high-order accuracy, with the following important improvements of the limiting procedure. First, the candidate polynomials of the p-weighted limiter are the p-hierarchical orthogonal polynomials of the current cell, and the linear polynomials constructed by minimizing the projection error on the face-neighboring cells. Second, the p-weighted procedure introduces a new smoothness indicator which has less numerical dissipation comparing with the classical WENO one. The smoothness indicator is efficiently computed through a quadrature-free approach that takes advantage of the orthogonal property of the basis functions. Third, the small positive number ϵ, which is introduced in the weights to avoid dividing by zero, is set as a function of the smoothness indicator to preserve accuracy near smooth extremas. Numerous benchmark problems are solved to test the p1, p3 and p5 discontinuous Galerkin schemes using the p-weighted limiter. Numerical results demonstrate that the p-weighted limiter is capable of capturing strong shocks while preserving accuracy in smooth regions.