Optimization-based meta-learning methods for few-shot one-class classification (FS-OCC) aim to fine-tune a meta-trained model to classify the positive and negative samples using only a few positive samples by adaptation. However, recent approaches primarily focus on adjusting existing meta-learning algorithms for FS-OCC, while overlooking issues stemming from the misalignment between the cross-entropy loss and OCC tasks during adaptation. This misalignment, combined with the limited availability of one-class samples and the restricted diversity of task-specific adaptation, can significantly exacerbate the adverse effects of gradient instability and generalization. To address these challenges, we propose a novel \textbf{T}ask-\textbf{S}pecific \textbf{G}radient \textbf{A}daptation (\textbf{TSGA}) for FS-OCC. Without extra supervision, TSGA learns to generate appropriate, stable gradients by leveraging label prediction and feature representation details of one-class samples and refines the adaptation process by recalibrating task-specific gradients and regularization terms. We evaluate TSGA on three challenging datasets and a real-world CNC Milling Machine application and demonstrate consistent improvements over baseline methods. Furthermore, we illustrate the critical impact of gradient instability and task-agnostic adaptation. Notably, TSGA achieves state-of-the-art results by effectively addressing these issues.