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 finergrained 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.