Strategic reasoning enables agents to cooperate, communicate, and compete with other agents in diverse situations. Existing approaches to solving strategic games rely on extensive training, yielding strategies that do not generalize to new scenarios or games without retraining. Large Language Models (LLMs), with their ability to comprehend and generate complex, context-rich language, could prove powerful as tools for strategic gameplay. This paper introduces an approach that uses pretrained LLMs with few-shot chain-of-thought examples to enable strategic reasoning for AI agents. Our approach uses systematically generated demonstrations of reasoning about states, values, and beliefs to prompt the model. Using extensive variations of simple matrix games, we show that strategies that are derived based on systematically generated prompts generalize almost perfectly to new game structures, alternate objectives, and hidden information. Additionally, we demonstrate our approach can lead to human-like negotiation strategies in realistic scenarios without any extra training or fine-tuning. Our results highlight the ability of LLMs, guided by systematic reasoning demonstrations, to adapt and excel in diverse strategic scenarios.