Despite the impressive performance of large language models (LLMs) pretrained on vast knowledge corpora, advancing their knowledge manipulation-the ability to effectively recall, reason, and transfer relevant knowledge-remains challenging. Existing methods mainly leverage Supervised Fine-Tuning (SFT) on labeled datasets to enhance LLMs' knowledge manipulation ability. However, we observe that SFT models still exhibit the known&incorrect phenomenon, where they explicitly possess relevant knowledge for a given question but fail to leverage it for correct answers. To address this challenge, we propose KALE (Knowledge-Aware LEarning)-a post-training framework that leverages knowledge graphs (KGs) to generate high-quality rationales and enhance LLMs' knowledge manipulation ability. Specifically, KALE first introduces a Knowledge-Induced (KI) data synthesis method that efficiently extracts multi-hop reasoning paths from KGs to generate high-quality rationales for question-answer pairs. Then, KALE employs a Knowledge-Aware (KA) fine-tuning paradigm that enhances knowledge manipulation by internalizing rationale-guided reasoning through minimizing the KL divergence between predictions with and without rationales. Extensive experiments on eight popular benchmarks across six different LLMs demonstrate the effectiveness of KALE, achieving accuracy improvements of up to 11.72% and an average of 4.18%.