Autoregressive models capture stochastic processes in which past realizations determine the generative distribution of new data; they arise naturally in a variety of industrial, biomedical, and financial settings. A key challenge when working with such data is to determine when the underlying generative model has changed, as this can offer insights into distinct operating regimes of the underlying system. This paper describes a novel dynamic programming approach to localizing changes in high-dimensional autoregressive processes and associated error rates that improve upon the prior state of the art. When the model parameters are piecewise constant over time and the corresponding process is piecewise stable, the proposed dynamic programming algorithm consistently localizes change points even as the dimensionality, the sparsity of the coefficient matrices, the temporal spacing between two consecutive change points, and the magnitude of the difference of two consecutive coefficient matrices are allowed to vary with the sample size. Furthermore, the accuracy of initial, coarse change point localization estimates can be boosted via a computationally-efficient refinement algorithm that provably improves the localization error rate. Finally, a comprehensive simulation experiments and a real data analysis are provided to show the numerical superiority of our proposed methods.