This technology includes monoclonal antibodies (mAb) that specifically and with high affinity bind the final complement components C3dg and C3d (subsequently referred to as C3d), which can be used to kill tumor cells that carry C3d on their cell surface. We show that tumor cells of patients treated with the therapeutic anti-CD20 mAb ofatumumab carry C3d on the cell surface and can bind and be killed by addition of anti-C3 mAbs. In contrast, further addition of more ofatumumab has only minimal effects. This invention is a novel therapeutic concept to enhance the efficacy of therapeutic antibodies and the specific mAbs to achieve this goal. In a "one-two punch" approach, administration of a first tumor antigen-specific mAb that kills some but not all tumor cells is followed by an anti-C3d mAb that targets the C3d deposited on the tumor cells.