With the accelerating digital transformation of modern society, numerous data center (DC) agents are connected to the distribution power networks (DPNs) via microgrid and engaging in fierce market competition. To address the asymmetric operational risks faced by each data center agent, particularly those arising from market volatility and equipment failures, a novel cooperative game-theoretic approach is proposed in this paper. Firstly, a cooperative operation framework for the microgrid-integrated data centers (MDCs) system is established from two dimensions: joint task allocation across MDCs on the computing side and energy sharing among MDCs on the power side. Moreover, an optimal operating model for MDCs is established, which integrates the task allocation model that takes into account the task processing capacity of MDCs. Then, a cooperative operation model for the MDCs system based on Nash game theory is developed, and a joint solution framework for task allocation and the cooperative operation model is designed. Finally, the proposed cooperative game-theoretic approach is validated in a test system. The results show that the proposed approach ensures the reliable operation of the DPN while avoiding asymmetric operation risks among MDCs. It enhances the stability and security of distributed data processing. Furthermore, the Nash game-theoretic model achieves a symmetric distribution of profits and risks across MDCs, eliminating individual biases and maximizing the overall benefits of the cooperative alliance.