Credit card fraud represents a $28.58 billion annual challenge globally, with traditional detection systems struggling to adapt to evolving fraud patterns while maintaining low false positive rates. This paper introduces a novel four-stage Large Language Model (LLM) chaining framework for credit card fraud detection that sequentially processes transaction preprocessing, behavioral analysis, risk assessment, and decision synthesis. Our approach leverages the reasoning capabilities of modern LLMs through carefully orchestrated prompt engineering and context management techniques. Experimental evaluation on acomprehensive dataset of 2.1 million transactions demonstrates significant performance improvements, achieving 94.7% accuracy with only 2.1% false positive rate compared to traditional methods. The system provides human-interpretable explanations addressing regulatory compliance requirements while demonstrating superior detection of sophisticated fraud patterns including account takeover scenarios and synthetic identity fraud. Our cost-benefit analysis reveals a 1,153% ROI despite higher computational requirements, making it economically viable for large-scale deployment.