Self-training large language models (LLMs) with generated reasoning paths has emerged as a promising approach to improve performance on complex reasoning tasks. However, most existing methods rely on correctness-based supervision, treating samples that reach the correct answer as high-quality despite potentially flawed intermediate steps, leading to noisy training signals. In this work, we propose K-STaR (Knowledge-aware Self-Taught Reasoner), a self-training framework that verifies reasoning paths through knowledge elicitation and integration as a proxy, without requiring any external reward models or dense step-by-step annotations. K-STaR models reasoning as a structured composition of knowledge units and automatically assigns process rewards to intermediate steps via consistency and frequency analysis, ensuring that only knowledge-grounded reasoning paths are retained. Experiments on mathematical and commonsense reasoning tasks show that K-STaR consistently discovers higher-quality reasoning paths and achieves superior self-training performance compared to prior methods. Our results highlight the importance of moving beyond correctness-centric supervision toward knowledge-grounded self-improvement.