Graph-level tasks on dynamic graphs are prevalent in real-world applications, yet methods for this problem remain limited. Existing models exhibit poor parallel processing capabilities and struggle to capture high-order subgraph evolution patterns. To bridge this gap, we propose the Dynamic Graph Transformer for Graph-level Classification (DGTC), which integrates a global self-attention mechanism to adaptively aggregate node information across the entire graph. This architecture eliminates the need for sequential processing of temporal edges and enables highly parallel computation. To comprehensively model the spatio-temporal evolution of dynamic graphs, we introduce (i) dynamic degree encoding to capture local structural variation; and (ii) dynamic distance encoding to track changes in higher-order subgraph structure via time-varying pairwise distances. Extensive experiments show that DGTC delivers an average performance gain of 14.28% and up to 35 \(\times \) speedup over ten competitive baselines. Code is available at https://github.com/wangz3066/DGTC .