Entropy regularization is a popular method in reinforcement learning (RL). Although it has many advantages, it alters the RL objective and makes the converged policy deviate from the optimal policy of the original Markov Decision Process (MDP). Though divergence regularization has been proposed to settle this problem, it cannot be trivially applied to cooperative multi-agent reinforcement learning (MARL). In this paper, we investigate divergence regularization in cooperative MARL and propose a novel off-policy cooperative MARL framework, divergence-regularized multi-agent actor-critic (DMAC). Theoretically, we derive the update rule of DMAC which is naturally off-policy, guarantees the monotonic policy improvement and convergence in both the original MDP and the divergence-regularized MDP, and is not biased by the regularization. We also give a bound of the discrepancy between the converged policy and the optimal policy in the original MDP. DMAC is a flexible framework and can be combined with many existing MARL algorithms. Empirically, we evaluate DMAC in a didactic stochastic game and StarCraft Multi-Agent Challenge and show that DMAC substantially improves the performance of existing MARL algorithms.