With the growing deployment of multi-server mobile edge computing (MEC) systems, efficient resource allocation becomes critical to minimizing task delay and ensuring service quality in dynamic environments. Traditional supervised deep learning-based resource allocation (SDLRA) methods, while effective, rely heavily on labeled training data and require costly retraining when network conditions change. To address these limitations, we propose a continual graph Learning-based resource allocation (CGLRA) method that integrates graph neural networks (GNN), unsupervised learning, and continual learning (CL). GNNs are employed to capture the topological dependencies among users and MEC servers, unsupervised learning eliminates the need for labeled data by directly optimizing system-level performance metrics, and the CL mechanism allows the model to adapt to new tasks while retaining knowledge from previously seen configurations, thus avoiding catastrophic forgetting. Extensive experiments confirm that CGLRA effectively reduces task delays without requiring labeled data, and maintains robust performance across dynamic MEC configurations without retraining.