Recent applications of pattern recognition techniques on brain connectome classification using functional connectivity (FC) are shifting towards acknowledging the non-Euclidean topology and dynamic aspects of brain connectivity across time. In this paper, a deep spatiotemporal variational Bayes (DSVB) framework is proposed to learn time-varying topological structures in dynamic FC networks for identifying autism spectrum disorder (ASD) in human participants. The framework incorporates a spatial-aware recurrent neural network with an attention-based message passing scheme to capture rich spatiotemporal patterns across dynamic FC networks. To overcome model overfitting on limited training datasets, an adversarial training strategy is introduced to learn graph embedding models that generalize well to unseen brain networks. Evaluation on the ABIDE resting-state functional magnetic resonance imaging dataset shows that our proposed framework substantially outperforms state-of-the-art methods in identifying patients with ASD. Dynamic FC analyses with DSVB-learned embeddings reveal apparent group differences between ASD and healthy controls in brain network connectivity patterns and switching dynamics of brain states.