We generalize the Bak-Sneppen model of coevolution to a game model for evolutionary dynamics which provides a natural way for the emergence of cooperation. Interaction between members is mimicked by a prisoner's dilemma game with a memoryless stochastic strategy. The fitness of each member is determined by the payoffs π of the games with its neighbors. We investigate the evolutionary dynamics using a mean-field calculation and Monte Carlo method with two types of death processes, fitness-dependent death and chain-reaction death. In the former, the death probability is proportional to e^{-βπ} where β is the "selection intensity." The neighbors of the death site also die with a probability R through the chain-reaction process invoked by the abrupt change of the interaction environment. When a cooperator interacts with defectors, the cooperator is likely to die due to its low payoff, but the neighboring defectors also tend to disappear through the chain-reaction death, giving rise to an assortment of cooperators. Owing to this assortment, cooperation can emerge for a wider range of R values than the mean-field theory predicts. We present the detailed evolutionary dynamics of our model and the conditions for the emergence of cooperation.