Audio classification is considered as a challenging problem in pattern recognition. Recently, many algorithms have been proposed using deep neural networks. In this paper, we introduce a new attention-based neural network architecture called Classifier-Attention-Based Convolutional Neural Network (CAB-CNN). The algorithm uses a newly designed architecture consisting of a list of simple classifiers and an attention mechanism as a classifier selector. This design significantly reduces the number of parameters required by the classifiers and thus their complexities. In this way, it becomes easier to train the classifiers and achieve a high and steady performance. Our claims are corroborated by the experimental results. Compared to the state-of-the-art algorithms, our algorithm achieves more than 10% improvements on all selected test scores.