Breast cancer (BC) is spreading more and more every day. Therefore, a patient's life can be saved by its early discovery. Mammography is frequently used to diagnose BC. The classification of mammography region of interest (ROI) patches (i.e., normal, malignant, or benign) is the most crucial phase in this process since it helps medical professionals to identify BC. In this paper, a hybrid technique that carries out a quick and precise classification that is appropriate for the BC diagnosis system is proposed and tested. Three different Deep Learning (DL) Convolution Neural Network (CNN) models—namely, Inception-V3, ResNet50, and AlexNet—are used in the current study as feature extractors. To extract useful features from each CNN model, our suggested method uses the Term Variance (TV) feature selection algorithm. The TV-selected features from each CNN model are combined and a further selection is performed to obtain the most useful features which are sent later to the multiclass support vector machine (MSVM) classifier. The Mammographic Image Analysis Society (MIAS) image database was used to test the effectiveness of the suggested method for classification. The mammogram's ROI is retrieved, and image patches are assigned to it. Based on the results of testing several TV feature subsets, the 600-feature subset with the highest classification performance was discovered. Higher classification accuracy (CA) is attained when compared to previously published work. The average CA for 70% of training is 97.81%, for 80% of training, it is 98%, and for 90% of training, it reaches its optimal value. Finally, the ablation analysis is performed to emphasize the role of the proposed network’s key parameters.