Rapid and massive adoption of mobile/ online payment services has brought new challenges to the service providers as well as regulators in safeguarding the proper uses such services/ systems. In this paper, we leverage recent advances in deep-neural-network-based graph representation learning to detect abnormal/ suspicious financial transactions in real-world e-payment networks. In particular, we propose an end-to-end Graph Convolution Network (GCN)-based algorithm to learn the embeddings of the nodes and edges of a large-scale time-evolving graph. In the context of e-payment transaction graphs, the resultant node and edge embeddings can effectively characterize the user-background as well as the financial transaction patterns of individual account holders. As such, we can use the graph embedding results to drive downstream graph mining tasks such as node-classification to identify illicit accounts within the payment networks. Our algorithm outperforms state-of-the-art schemes including GraphSAGE, Gradient Boosting Decision Tree and Random Forest to deliver considerably higher accuracy (94.62% and 86.98% respectively) in classifying user accounts within 2 practical e-payment transaction datasets. It also achieves outstanding accuracy (97.43%) for another biomedical entity identification task while using only edge-related information.