In this paper, we study the structural and electronic properties of graphene adsorbed on MoS2 monolayer (G/MoS2) with different stacking configurations using dispersion-corrected density functional theory. Our calculations show that the interaction between graphene and MoS2 monolayer is a weak van der Waals interaction in all four stacking configurations with the binding energy per carbon atom of −30 meV. In the presence of MoS2 monolayer, the linear bands on the Dirac cone of graphene at the interfaces are slightly split. A band gap about 3 meV opens in G/MoS2 interfaces due to the breaking of sublattice symmetry by the intrinsic interface dipole, and it could be effectively modulated by the stacking configurations. Furthermore, we found that an n-type Schottky contact is formed at the G/MoS2 interface in all four stacking configurations with a small Schottky barrier about 0.49 eV. The appearance of the non-zero band gap in graphene has opened up new possibilities for its application in electronic devices such as graphene field-effect transistors.