Fraud detection in payment, e-commerce, and telecommunications systems requires accuracy at the individual level, robustness under severe class imbalance, and ease of understanding for risk managers. Existing methods fall at least one of these requirements: automated machine learning systems search a fixed numerical space without semantic awareness of the dataset; graph neural network-based methods require pre-defined relational graphs and remain opaque at the individual-decision level; and the design of general-purpose large language model (LLM) agents does not consider the recall and precision constraints specific to real-world fraud detection. In this paper, we propose SAGE, the first end-to-end LLM-driven multi-agent framework for fraud detection. SAGE coordinates three dedicated agents that make decisions based on a six-layer Data Diagnostic Tree (DDT) and a Markov decision process guided by natural-language gradients, automatically optimizing the model under a fraud-specific reward. On five fraud datasets and five LLM backbones, SAGE wins $96.00\%$ of method--dataset comparisons and improves F1 by an average of $40.86\%$ over baselines. The code is available at https://github.com/yichenC1c/SAGE.