This paper studies the problem of class-imbalanced graph classification, which aims at effectively classifying the graph categories in scenarios with imbalanced class distributions. While graph neural networks (GNNs) have achieved remarkable success, their modeling ability on imbalanced graph-structured data remains suboptimal, which typically leads to predictions biased towards the majority classes. On the other hand, existing class-imbalanced learning methods in vision may overlook the rich graph semantic substructures of the majority classes and excessively emphasize learning from the minority classes. To address these challenges, we propose a simple yet powerful approach called C$^3$GNN that integrates the idea of clustering into contrastive learning to enhance class-imbalanced graph classification. Technically, C$^3$GNN clusters graphs from each majority class into multiple subclasses, with sizes comparable to the minority class, mitigating class imbalance. It also employs the Mixup technique to generate synthetic samples, enriching the semantic diversity of each subclass. Furthermore, supervised contrastive learning is used to hierarchically learn effective graph representations, enabling the model to thoroughly explore semantic substructures in majority classes while avoiding excessive focus on minority classes. Extensive experiments on real-world graph benchmark datasets verify the superior performance of our proposed method against competitive baselines.