Static Application Security Testing (SAST) tools are critical to software quality, identifying potential code issues early in development. However, they often produce false positive warnings that require manual review, slowing down development. Thus, automating false positive mitigation (FPM) is essential. The advent of Large Language Models (LLMs), with their strong abilities in natural language and code understanding, offers promising avenues for FPM. Yet current LLM-based FPM method faces two major limitations: 1. The warning-related code snippets extracted are overly broad and cluttered with irrelevant control/data flows, reducing precision; 2. Critical code contexts are missing, leading to incomplete representations that can mislead LLMs and cause inaccurate assessments. To overcome these limitations, we propose LLM4FPM , a lightweight and efficient false positive mitigation framework. It features eCPG-Slicer, which builds an extended code property graph (eCPG) to extract precise line-level code contexts for warnings. Furthermore, the integrated FARF algorithm builds a file reference graph to identify all files that are relevant to warnings in linear time. This enables eCPG-Slicer to obtain rich contextual information without resorting to expensive whole-program analysis. LLM4FPM outperforms the existing method on the Juliet dataset (F1 > 99% across various Common Weakness Enumerations) and improves label accuracy on the D2A dataset to 86%. By leveraging a lightweight open-source LLM, LLM4FPM can significantly save inspection costs up to \$2758 per run (\$0.384 per warning) on Juliet with an average inspection time of 4.7s per warning. Moreover, real-world tests on popular C/C++ projects demonstrate its practicality.