Many endeavors have sought to develop countermeasure techniques as enhancements on Automatic Speaker Verification (ASV) systems, in order to make them more robust against spoof attacks. As evidenced by the latest ASVspoof 2019 countermeasure challenge, models currently deployed for the task of ASV are, at their best, devoid of suitable degrees of generalization to unseen attacks. A joint improvement of components of ASV spoof detection systems including the classifier, feature extraction phase, and model loss function may lead to a better detection of attacks by these systems. Accordingly, the present study proposes the Efficient Attention Branch Network (EABN) architecture with a combined loss function to address the model generalization to unseen attacks. The EABN is based on attention and perception branches. The attention branch provides an attention mask that improves the classification performance and at the same time is interpretable from a human point of view. The perception branch, is used for our main purpose which is spoof detection. The new EfficientNet-A0 architecture was optimized and employed for the perception branch, with nearly ten times fewer parameters and approximately seven times fewer floating-point operations than the SE-Res2Net50 as the best existing network. The proposed method on ASVspoof 2019 dataset achieved EER = 0.86% and t-DCF = 0.0239 in the Physical Access (PA) scenario using the logPowSpec as the input feature extraction method. Furthermore, using the LFCC feature, and the SE-Res2Net50 for the perception branch, the proposed model achieved EER = 1.89% and t-DCF = 0.507 in the Logical Access (LA) scenario, which to the best of our knowledge, is the best single system ASV spoofing countermeasure method.