Graph neural networks (GNNs) have shown superior performance in node classification. However, GNNs perform poorly in the few-shot node classification (FSNC) task that requires robust generalization to make accurate predictions for unseen classes with limited labels. To tackle the challenge, we propose the integration of sharpness-aware minimization (SAM)---a technique designed to enhance model generalization by finding a flat minimum of the loss landscape---into GNN training. The standard SAM approach, however, consists of two forward-backward steps in each training iteration, doubling the computational cost compared to the base optimizer (e.g., Adam). To mitigate this drawback, we introduce a novel algorithm, Fast Graph Sharpness-Aware Minimization (FGSAM), that integrates the rapid training of multilayer perceptrons (MLPs) with the superior performance of GNNs. Specifically, we utilize GNNs for parameter perturbation while employing MLPs to minimize the perturbed loss so that we can find a flat minimum with good generalization more efficiently. Moreover, our method ingeniously reutilizes the gradient from the perturbation phase to incorporate graph topology into the minimization process at almost zero additional cost. To further enhance training efficiency, we develop FGSAM+ that executes exact perturbations periodically. Extensive experiments demonstrate that our proposed algorithm outperforms the standard SAM with lower computational costs in FSNC tasks. In particular, our FGSAM+ as a SAM variant, for the first time, offers a faster optimization than the base optimizer in most cases. In addition to FSNC, our proposed methods also demonstrate competitive performance in the standard node classification task for heterophilic graphs, highlighting the broad applicability.