Evolutionary Game Theory is an application of game theory to evolving populations of organisms. Of recent interest are EGT models situated on structured populations or spatial evolutionary games. Due to the complexity added by introducing a population structure, model analysis is usually performed through agent-based Monte-Carlo simulations. However, it can be difficult to obtain desired quantities of interest from these simulations due to stochastic effects. We define a framework for modeling spatial evolutionary games using Dynamic Bayesian Networks that capture the underlying stochastic process. The resulting Dynamic Bayesian Networks can be queried for quantities of interest by performing exact inference on the network. We then propose a method for producing approximations of the spatial evolutionary game through the truncation of the corresponding DBN, taking advantage of the high symmetry of the model. This method generalizes mean-field and pair approximations in the literature for spatial evolutionary games. Furthermore, we show empirical results demonstrating the capability of the method to obtain much better accuracy than pair approximation with respect to stochastic simulations.