We investigate the electronic transport properties of copper-graphene composites using a density-functional framework. Conduction in composites by varying the interface distance of a copper/graphene/copper (Cu/G/Cu) interface models was studied. The electronic density of states reveals increasing contributions from both copper and carbon atoms near the Fermi level with decreasing Cu-G interfacial distance. Electronic conductivity of the models computed using the Kubo-Greenwood formula showed the conductivity increases with decreasing Cu-G distance. We also find that the conductivity saturates below a threshold Cu-G distance. By computing the space-projected conductivity of the Cu/G/Cu models, we show that the graphene forms a bridge to the electronic conduction at small copper-graphene distances, thereby enhancing the conductivity.