Lithology classification aims to infer subsurface rock types from well-logging signals, supporting downstream applications like reservoir characterization. Despite substantial progress, most existing methods still treat lithology classification as a single-pass classification task. In contrast, practical experts incorporate geological principles, external knowledge, and tool-use capabilities to perform accurate classification. In this work, we propose GeoDecider, a coarse-to-fine agentic workflow that enables accurate and explainable lithology classification through training-free use of large language models (LLMs). GeoDecider reformulates lithology classification as an expert-like structured process and organizes it into a multi-stage workflow involving coarse-to-fine reasoning. Specifically, GeoDecider includes the following stages: (1) base classifier-guided coarse classification, which uses a pre-trained classifier to provide a rough reference for downstream tasks, thus reducing the overall cost of downstream reasoning, (2) tool-augmented reasoning, which utilizes several tools such as contextual analysis and neighbor retrieval to achieve finer and more precise classifications, (3) geological refinement, which post-processes the final results to enforce geological consistency. Experiments on four benchmarks show that GeoDecider outperforms representative baselines. Further analysis demonstrates that the proposed framework produces geologically interpretable predictions while achieving a better trade-off between classification performance and inference efficiency.