Patient matching is the process of linking patients to appropriate clinical trials by accurately identifying and matching their medical records with trial eligibility criteria. We propose LLM-Match, a novel framework for patient matching leveraging fine-tuned open-source large language models. Our approach consists of four key components. First, a retrieval-augmented generation (RAG) module extracts relevant patient context from a vast pool of electronic health records (EHRs). Second, a prompt generation module constructs input prompts by integrating trial eligibility criteria (both inclusion and exclusion criteria), patient context, and system instructions. Third, a fine-tuning module with a classification head optimizes the model parameters using structured prompts and ground-truth labels. Fourth, an evaluation module assesses the fine-tuned model's performance on the testing datasets. We evaluated LLM-Match on four open datasets, n2c2, SIGIR, TREC 2021, and TREC 2022, using open-source models, comparing it against TrialGPT, Zero-Shot, and GPT-4-based closed models. LLM-Match outperformed all baselines.