Retrieval-augmented generation (RAG) systems enhance large language models (LLMs) by integrating external knowledge bases, but this advancement introduces significant privacy risks. Existing privacy attacks on RAG systems can trigger data leakage but often fail to accurately isolate knowledge-base-derived sentences within mixed responses. They also lack robustness when applied across multiple domains. This paper addresses these challenges by presenting a novel black-box attack framework that exploits knowledge asymmetry between RAG and standard LLMs to achieve fine-grained privacy extraction across heterogeneous knowledge landscapes. We propose a chain-of-thought reasoning strategy that creates adaptive prompts to steer RAG systems away from sensitive content. Specifically, we first decompose adversarial queries to maximize information disparity and then apply a semantic relationship scoring to resolve lexical and syntactic ambiguities. We finally train a neural network on these feature scores to precisely identify sentences containing private information. Unlike prior work, our framework generalizes to unseen domains through iterative refinement without pre-defined knowledge. Experimental results show that we achieve over 91% privacy extraction rate in single-domain and 83% in multi-domain scenarios, reducing sensitive sentence exposure by over 65% in case studies. This work bridges the gap between attack and defense in RAG systems, enabling precise extraction of private information while providing a foundation for adaptive mitigation.