In contemporary scientific research, it is of great interest to predict a categorical response based on a high-dimensional tensor and additional covariates. Motivated by various applications, we propose a comprehensive and interpretable discriminant analysis model, called CATCH (Covariate-Adjusted Tensor Classification in High-dimensions), which efficiently integrates the covariates and the tensor to predict the categorical outcome. The CATCH model jointly models the relationships among the covariates, the tensor predictor, and the categorical response. It preserves and utilizes the intrinsic structure of the data for maximum interpretability and optimal prediction. To tackle the challenges arising from the tensor dimensions, we propose a group penalized approach to select a subset of tensor predictor entries that has direct discriminative effect after adjusting for covariates. We further develop an efficient algorithm that takes advantage of the tensor structure. Theoretical results confirm that our method achieves variable selection consistency and optimal prediction. The superior performance over existing methods is demonstrated in extensive simulation and real data studies.