Static analysis tools are widely used to detect bugs, vulnerabilities, and code smells. Traditionally, developers must resolve these warnings manually. Because this process is tedious, developers sometimes ignore warnings, leading to an accumulation of warnings and a degradation of code quality. This paper presents CodeCureAgent, an approach that harnesses LLM-based agents to automatically analyze, classify, and repair static analysis warnings. Unlike previous work, our method does not follow a predetermined algorithm. Instead, we adopt an agentic framework that iteratively invokes tools to gather additional information from the codebase (e.g., via code search) and edit the codebase to resolve the warning. CodeCureAgent detects and suppresses false positives, while fixing true positives when identified. We equip CodeCureAgent with a three-step heuristic to approve patches: (1) build the project, (2) verify that the warning disappears without introducing new warnings, and (3) run the test suite. We evaluate CodeCureAgent on a dataset of 1,000 SonarQube warnings found in 106 Java projects and covering 291 distinct rules. Our approach produces plausible fixes for 96.8% of the warnings, outperforming state-of-the-art baseline approaches by 29.2%-34.0% in plausible-fix rate. Manual inspection of 291 cases reveals a correct-fix rate of 86.3%, showing that CodeCureAgent can reliably repair static analysis warnings. The approach incurs LLM costs of about 2.9 cents (USD) and an end-to-end processing time of about four minutes per warning. We envision CodeCureAgent helping to clean existing codebases and being integrated into CI/CD pipelines to prevent the accumulation of static analysis warnings.