Open-domain dialogue systems need to grasp social commonsense to understand and respond effectively to human users. Commonsense-augmented dialogue models have been proposed that aim to infer commonsense knowledge from dialogue contexts in order to improve response quality. However, existing approaches to commonsense-augmented dialogue rely on implicit reasoning to integrate commonsense inferences during response generation. In this study, we explore the impact of explicit reasoning against implicit reasoning over commonsense for dialogue response generation. Our findings demonstrate that separating commonsense reasoning into explicit steps for generating, selecting, and integrating commonsense into responses leads to better dialogue interactions, improving naturalness, engagement, specificity, and overall quality. Subsequent analyses of these findings unveil insights into the effectiveness of various types of commonsense in generating responses and the particular response traits enhanced through explicit reasoning for commonsense integration. Our work advances research in open-domain dialogue by achieving a new state-of-the-art in commonsense-augmented response generation.