To improve device robustness, a highly desirable key feature of a competitive data-driven acoustic scene classification (ASC) system, a novel two-stage system based on fully convolutional neural networks (CNNs) is proposed. Our two-stage system leverages on an ad-hoc score combination based on two CNN classifiers: (i) the first CNN classifies acoustic inputs into one of three broad classes, and (ii) the second CNN classifies the same inputs into one of ten finer-grained classes. Three different CNN architectures are explored to implement the two-stage classifiers, and a frequency sub-sampling scheme is investigated. Moreover, novel data augmentation schemes for ASC are also investigated. Evaluated on DCASE 2020 Task 1a, our results show that the proposed ASC system attains a state-of-the-art accuracy on the development set, where our best system, a two-stage fusion of CNN ensembles, delivers a 81.9% average accuracy among multi-device test data, and it obtains a significant improvement on unseen devices. Finally, neural saliency analysis with class activation mapping (CAM) gives new insights on the patterns learnt by our models.