Corporate social responsibility (CSR) is widely noticed as an essential tool for business operation and sustainable development. Meanwhile, the fiercely competitive external environment and unpredictable events prompt enterprises’ cooperation to prevent supply chain collapse. We investigate the cooperative strategy in a live-streaming supply chain (LSC) consisting of a dominant brand owner, a retailer, and a live streamer, where the brand owner considers CSR by considering the welfare of stakeholders. We construct one non-cooperative and three cooperative Stackelberg game models to explore the impact of CSR on cooperative strategy and LSC operations. The results show the following. (1) When the brand owner considers CSR, LSC members and systems are more profitable in the four models than when the brand owner does not consider CSR. (2) When the flow effect is small, the brand owner tends to cooperate with the retailer; otherwise, the brand owner prefers to cooperate with the live streamer. (3) The grand coalition C (the brand owner cooperates with the retailer and live streamer) is the consistent strategy for the LSC system, consumers, and society. These findings help enterprises recognize the importance of CSR and collaboration, thus further providing reference opinions on engaging in CSR and how to achieve collaboration.