Software vulnerabilities pose serious risks to modern software ecosystems. While the National Vulnerability Database (NVD) is the authoritative source for cataloging these vulnerabilities, it often lacks explicit links to the corresponding Vulnerability-Fixing Commits (VFCs). VFCs encode precise code changes, enabling vulnerability localization, patch analysis, and dataset construction. Automatically mapping NVD records to their true VFCs is therefore critical. Existing approaches have limitations as they rely on sparse, often noisy commit messages and fail to capture the deep semantics in the vulnerability descriptions. To address this gap, we introduce PatchSeeker, a novel method that leverages large language models to create rich semantic links between vulnerability descriptions and their VFCs. PatchSeeker generates embeddings from NVD descriptions and enhances commit messages by synthesizing detailed summaries for those that are short or uninformative. These generated messages act as a semantic bridge, effectively closing the information gap between natural language reports and low-level code changes. Our approach PatchSeeker achieves 59.3% higher MRR and 27.9% higher Recall@10 than the best-performing baseline, Prospector, on the benchmark dataset. The extended evaluation on recent CVEs further confirms PatchSeeker's effectiveness. Ablation study shows that both the commit message generation method and the selection of backbone LLMs make a positive contribution to PatchSeeker. We also discuss limitations and open challenges to guide future work.