Despite recent advances in the accuracy of brain tumor segmentation, the results still suffer from low reliability and robustness. Uncertainty estimation is an efficient solution to this problem, as it provides a measure of confidence in the segmentation results. The current uncertainty estimation methods based on quantile regression, Bayesian neural network, ensemble, and Monte Carlo dropout are limited by their high computational cost and inconsistency. In order to overcome these challenges, Evidential Deep Learning (EDL) was developed in recent work but primarily for natural image classification and showed inferior segmentation results. In this paper, we proposed a region-based EDL segmentation framework that can generate reliable uncertainty maps and accurate segmentation results, which is robust to noise and image corruption. We used the Theory of Evidence to interpret the output of a neural network as evidence values gathered from input features. Following Subjective Logic, evidence was parameterized as a Dirichlet distribution, and predicted probabilities were treated as subjective opinions. To evaluate the performance of our model on segmentation and uncertainty estimation, we conducted quantitative and qualitative experiments on the BraTS 2020 dataset. The results demonstrated the top performance of the proposed method in quantifying segmentation uncertainty and robustly segmenting tumors. Furthermore, our proposed new framework maintained the advantages of low computational cost and easy implementation and showed the potential for clinical application.