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The high cost of high-resolution computational fluid/flame dynamics (CFD) has hindered its application in combustion related research, design and optimization. In this study, we propose a framework for turbulent combustion simulation based on the deep learning approach. An optimized deep convolutional neural network (CNN) inspired from a U-Net architecture and inception module is designed for constructing the deep learning solver, named CFDNN. CFDNN is then trained on simulation results of hydrogen combustion in a cavity with different inlet velocities. After training, CFDNN can not only accurately predict the flow and combustion fields within the range of the training set, but also shows an extrapolation ability for prediction outside the training set. The results from the CFDNN solver show excellent consistency with the conventional CFD results in terms of both predicted spatial distributions and temporal dynamics. Meanwhile, two orders of magnitude of acceleration are achieved by using the CFDNN solver compared to a conventional CFD solver. The successful development of such a deep learning-based solver opens up new possibilities of low-cost, high-accuracy simulations, fast prototyping, design optimization and real-time control of combustion systems such as gas turbines and scramjets.