Rock elastic properties play an important role in the geological characteristics of reservoirs. The analysis of these properties is normally based physical experiments on rocks. However, such conventional physical experiments cannot meet actual requirements when the rock is fragile or has complex composition. With the development of computer technology and the application of micro-computed tomography scanning technology, digital rock physics technologies came into existence. In this work, micro-computed tomography was applied to obtain high-quality three-dimensional images of coal samples. Next, the image multi-threshold segmentation method was used to divide the grayscale image into three reasonable components, including mineral, organic matrix, and pores. Digital rock models with different gas saturations were established using mathematical morphology based methods. Five volume samples were selected from the original large digital rock model under different conditions of porosity, mineral, and gas saturation. Based on these three-dimensional digital cores and the finite element method, the effective elastic moduli of coal rock mass were simulated and the compressional wave velocity and shear wave velocity were computed. Results show that, in the absence of filled minerals, both bulk and shear moduli decrease with rising porosity; compressional and shear wave velocities decline, and the ratio of compressional wave velocity to shear wave velocity increases. However, a more realistic study considering filled minerals demonstrates decreasing shear wave velocity and counterintuitively rising compressional wave velocity when the porosity increases. Gas saturation only affects the compressional wave velocity. The obtained results improve our understanding of rock elastic behaviors in the coalbed.