Large Vision-Language Models (LVLMs) face a fundamental dilemma in video reasoning: they are caught between the prohibitive computational costs of verbose reasoning and the hallucination risks of efficient, ungrounded approaches. To resolve this, we introduce the Chain of Evidence (CoE), a novel framework that architecturally decouples and co-optimizes perceptual grounding and reasoning efficiency. CoE incorporates two core innovations: (1) A lightweight Evidence Grounding Module (EGM) that acts as a query-guided filter, dynamically identifying and extracting a compact set of high-fidelity visual evidence; and (2) An Evidence-Anchoring Protocol optimized via Reinforcement Learning. Crucially, we design a composite reward mechanism that enforces process alignment, compelling the model to strictly reference identified temporal anchors during deduction, thereby mitigating hallucinations. To enable this, we construct CoE-Instruct, a large-scale dataset (164k samples) featuring a novel dual-annotation schema for separate perception and reasoning supervision. Extensive experiments on five benchmarks, including Video-MME, MVBench, and VSI-Bench, demonstrate that CoE-enhanced models establish a new state-of-the-art. They significantly outperform existing methods in accuracy, proving CoE to be a powerful and practical paradigm for reliable video understanding.