Temporal graph classification is an emerging task with broad applications in neuroscience, cybersecurity, bioinformatics, and infrastructure monitoring, where systems are naturally modeled as evolving networks. While recent advances in temporal graph neural networks (TGNNs) have enabled the modeling of dynamic graph data, they often struggle to capture global structural evolution, suffer from oversmoothing, and are sensitive to noise and node permutations. In this work, we propose a novel framework that integrates tools from topological data analysis (TDA) into the temporal graph classification pipeline to address these limitations. By extracting persistent topological features from time-evolving graphs, our method captures stable, global structural patterns such as cycles and connectivity changes that are difficult for standard TGNNs to learn. These topological descriptors are then combined with neural architectures to enhance representation learning and improve classification performance. We evaluate our approach on multiple real-world datasets and demonstrate that it consistently outperforms existing TGNN models, particularly in tasks where the structural dynamics of the graph are critical to the target labels. Our results highlight the potential of topological machine learning in enriching temporal graph models with geometric and structural priors.