In recent years, illicit activities in cryptocurrency transactions have posed a severe threat to financial systems and society. Current detection methods mainly rely on machine learning and graph neural networks but often overlook the dynamic evolution of transaction graphs over time. To address this, we propose a novel Evolutionary Graph Self-Attention Network (EG-SAN) that can simultaneously capture the spatial relationships among nodes as well as their temporal evolution features through a joint self-attention mechanism across structural and temporal dimensions. Specifically, the graph attention network aggregates features from node neighborhoods in the graph data, while gated recurrent units dynamically update its layer weights to capture the evolution of the graph over the temporal dimension. Subsequently, a self-attention mechanism further optimizes these weights, enabling the model to more accurately reflect the salient features of the data at different time steps. We evaluate EG-SAN on a Bitcoin transaction dataset, and the results demonstrate its superiority over existing state-of-the-art methods. Further application to graph datasets from other domains also confirms the effectiveness and robustness of EG-SAN, showcasing its broad application potential.