Organic pollutants in industrial and agricultural sewage are a serious threat to the environment and human health. Achieving continuous photocatalytic degradation of organic pollutants under light and dark conditions would have exciting implications for practical sewage treatment. In this paper, CuS/g-C3N4 composite catalysts with CuS nanoparticles anchored on g-C3N4 sheets were successfully fabricated via a simple solvothermal reaction. The morphology, structure, optical absorption characteristics, electron–hole recombination rate, and degradation performance of the as-prepared CuS/g-C3N4 catalysts were investigated in detail. The results confirmed that the as-fabricated CuS/g-C3N4 catalysts exhibited high Fenton-like catalytic degradation efficiencies in the dark, and rapid concerted Fenton-like catalytic, direct H2O2 photocatalytic and CuS/g-C3N4 photocatalytic degradation activities under visible light. Thus, the as-fabricated CuS/g-C3N4 catalysts can degrade organic pollutants continuously during both day and night. These degradation properties, along with the simple catalyst fabrication process, will facilitate the practical application of this system in the continuous removal of organic pollutants.