Financial news sentiment analysis is crucial for anticipating market movements. With the rise of AI techniques such as Large Language Models (LLMs), which demonstrate strong text understanding capabilities, there has been renewed interest in enhancing these systems. Existing methods, however, often struggle to capture the complex economic context of news and lack transparent reasoning, which undermines their reliability. We propose Analogy-Driven Financial Chain-of-Thought (AD-FCoT), a prompting framework that integrates analogical reasoning with chain-of-thought (CoT) prompting for sentiment prediction on historical financial news. AD-FCoT guides LLMs to draw parallels between new events and relevant historical scenarios with known outcomes, embedding these analogies into a structured, step-by-step reasoning chain. To our knowledge, this is among the first approaches to explicitly combine analogical examples with CoT reasoning in finance. Operating purely through prompting, AD-FCoT requires no additional training data or fine-tuning and leverages the model's internal financial knowledge to generate rationales that mirror human analytical reasoning. Experiments on thousands of news articles show that AD-FCoT outperforms strong baselines in sentiment classification accuracy and achieves substantially higher correlation with market returns. Its generated explanations also align with domain expertise, providing interpretable insights suitable for real-world financial analysis.