Breast cancer accounts for 30% of all female cancers. Accurately distinguishing dangerous malignant tumors from benign harmless ones is key to ensuring patients receive lifesaving treatments on time. However, as doctors currently do not identify 10% to 30% of breast cancers during regular assessment, automated methods to detect malignant tumors are desirable. Although several computerized methods for breast cancer classification have been proposed, convolutional neural networks (CNNs) have demonstrably outperformed other approaches. In this paper, we propose an automated method for the binary classification of breast cancer tumors as either malignant or benign that utilizes a bag of deep multi-resolution convolutional features (BoDMCF) extracted from histopathological images at four resolutions (40×, 100×, 200× and 400×) by three pre-trained state-of-the-art deep CNN models: ResNet-50, EfficientNetb0, and Inception-v3. The BoDMCF extracted by the pre-trained CNNs were pooled using global average pooling and classified using the support vector machine (SVM) classifier. While some prior work has utilized CNNs for breast cancer classification, they did not explore using CNNs to extract and pool a bag of deep multi-resolution features. Other prior work utilized CNNs for deep multi-resolution feature extraction from chest X-ray radiographs to detect other conditions such as pneumoconiosis but not for breast cancer detection from histopathological images. In rigorous evaluation experiments, our deep BoDMCF feature approach with global pooling achieved an average accuracy of 99.92%, sensitivity of 0.9987, specificity (or recall) of 0.9797, positive prediction value (PPV) or precision of 0.99870, F1-Score of 0.9987, MCC of 0.9980, Kappa of 0.8368, and AUC of 0.9990 on the publicly available BreaKHis breast cancer image dataset. The proposed approach outperforms the prior state of the art for histopathological breast cancer classification as well as a comprehensive set of CNN baselines, including ResNet18, InceptionV3, DenseNet201, EfficientNetb0, SqueezeNet, and ShuffleNet, when classifying images at any individual resolutions (40×, 100×, 200× or 400×) or when SVM is used to classify a BoDMCF extracted using any single pre-trained CNN model. We also demonstrate through a carefully constructed set of experiments that each component of our approach contributes non-trivially to its superior performance including transfer learning (pre-training and fine-tuning), deep feature extraction at multiple resolutions, global pooling of deep multiresolution features into a powerful BoDMCF representation, and classification using SVM.