Despite the impressive performance of general large language models(LLMs), many of their applications in specific domains (e.g., low-data and knowledge-intensive) still confront significant challenges. Supervised fine-tuning (SFT)---where a general LLM is further trained on a small labeled dataset to adapt for specific tasks or domains---has shown great power for developing domain-specific LLMs. However, existing SFT data primarily consist of Question and Answer (Q&A) pairs, which poses a significant challenge for LLMs to comprehend the correlation and logic of knowledge underlying the Q&A. To address this challenge, we propose a conceptually flexible and general framework to boost SFT, namely Knowledge Graph-Driven Supervised Fine-Tuning (KG-SFT). The key idea of KG-SFT is to generate high-quality explanations for each Q&A pair via a structured knowledge graph to enhance the knowledge comprehension and manipulation of LLMs. Specifically, KG-SFT consists of three components: Extractor, Generator, and Detector. For a given Q&A pair, (i) Extractor first identifies entities within Q&A pairs and extracts relevant reasoning subgraphs from external KGs, (ii) Generator then produces corresponding fluent explanations utilizing these reasoning subgraphs, and (iii) finally, Detector performs sentence-level knowledge conflicts detection on these explanations to guarantee the reliability. KG-SFT focuses on generating high-quality explanations to improve the quality of Q&A pair, which reveals a promising direction for supplementing existing data augmentation methods. Extensive experiments on fifteen different domains and six different languages demonstrate the effectiveness of KG-SFT, leading to an accuracy improvement of up to 18% and an average of 8.7% in low-data scenarios.