Digital breast tomosynthesis (DBT) is a 3D breast cancer screening technique that can overcome the limitations of standard 2D digital mammography. However, DBT images often suffer from artifacts stemming from acquisition conditions, a limited angular range, and low radiation doses. These artifacts have the potential to degrade the performance of automated breast tumor classification tools. Notably, most existing automated breast tumor classification methods do not consider the effect of DBT image quality when designing the classification models. In contrast, this paper introduces a novel deep learning-based framework for classifying breast tumors in DBT images. This framework combines global image quality-aware features with tumor texture descriptors. The proposed approach employs a two-branch model: in the top branch, a deep convolutional neural network (CNN) model is trained to extract robust features from the region of interest that includes the tumor. In the bottom branch, a deep learning model named TomoQA is trained to extract global image quality-aware features from input DBT images. The quality-aware features and the tumor descriptors are then combined and fed into a fully-connected layer to classify breast tumors as benign or malignant. The unique advantage of this model is the combination of DBT image quality-aware features with tumor texture descriptors, which helps accurately classify breast tumors as benign or malignant. Experimental results on a publicly available DBT image dataset demonstrate that the proposed framework achieves superior breast tumor classification results, outperforming all existing deep learning-based methods.