Excellent current-carrying tribological properties including the low-friction, high anti-wear, high current-carrying efficiency, and stability are important for the current-carrying application in transmitting electrical signals and power. Here, the Cu/graphene composites with graphene uniformly distributed in Cu matrix were successfully prepared by combining the electroless plating process and powder metallurgy process. The current-carrying tribological properties including friction, wear, and electrical stability of the Cu/graphene composites with brass pairs were investigated by varying normal applied load and sliding speed under multiple applied voltages. The friction reduction and anti-wear properties of Cu/graphene composites were enhanced by the introduction of graphene. The friction coefficient of the Cu/graphene composites keeps stable under current-carrying and non-current-carrying conditions due to the benefit of the graphene enhancement to Cu. The graphene on wear surface reduces friction force and wear. The current-carrying efficiency and stability increased with the increase of applied load but decreased with increasing sliding speed. The contact stability increased with applied load, while high sliding speed caused the drastic vibration of sliding contact. The studies can provide a beneficial guideline for the current-carrying applications of Cu/graphene composites to reduce the friction and wear.