Graph-based learning excels at capturing interaction patterns in diverse domains like recommendation, fraud detection, and particle physics. However, its performance often degrades under distribution shifts, especially those altering network connectivity. Current methods to address these shifts typically require retraining with the source dataset, which is often infeasible due to computational or privacy limitations. We introduce Test-Time Structural Alignment (TSA), a novel algorithm for Graph Test-Time Adaptation (GTTA) that adapts a pretrained model to align graph structures during inference without the cost of retraining. Grounded in a theoretical understanding of graph data distribution shifts, TSA employs three synergistic strategies: uncertainty-aware neighborhood weighting to accommodate neighbor label distribution shifts, adaptive balancing of self-node and aggregated neighborhood representations based on their signal-to-noise ratio, and decision boundary refinement to correct residual label and feature shifts. Extensive experiments on synthetic and real-world datasets demonstrate TSA's consistent outperformance of both non-graph TTA methods and state-of-the-art GTTA baselines.