Pathology image classification plays a crucial role in accurate medical diagnosis and treatment planning. Training high-performance models for this task typically requires large-scale annotated datasets, which are both expensive and time-consuming to acquire. Active Learning (AL) offers a solution by iteratively selecting the most informative samples for annotation, thereby reducing the labeling effort. However, most AL methods are designed under the assumption of a closed-set scenario, where all the unannotated images belong to target classes. In real-world clinical environments, the unlabeled pool often contains a substantial amount of Out-Of-Distribution (OOD) data, leading to low efficiency of annotation in traditional AL methods. Furthermore, most existing AL methods start with random selection in the first query round, leading to a significant waste of labeling costs in open-set scenarios. To address these challenges, we propose OpenPath, a novel open-set active learning approach for pathological image classification leveraging a pre-trained Vision-Language Model (VLM). In the first query, we propose task-specific prompts that combine target and relevant non-target class prompts to effectively select In-Distribution (ID) and informative samples from the unlabeled pool. In subsequent queries, Diverse Informative ID Sampling (DIS) that includes Prototype-based ID candidate Selection (PIS) and Entropy-Guided Stochastic Sampling (EGSS) is proposed to ensure both purity and informativeness in a query, avoiding the selection of OOD samples. Experiments on two public pathology image datasets show that OpenPath significantly enhances the model's performance due to its high purity of selected samples, and outperforms several state-of-the-art open-set AL methods. The code is available at \href{https://github.com/HiLab-git/OpenPath}{https://github.com/HiLab-git/OpenPath}..