We study energy-transport efficiency in light-harvesting planar and 3D complexes of two-level atomic quantum systems, embedded in a common thermal black-body radiation. We show that the collective non-local dissipation induced by the thermal bath plays a fundamental role in energy transport. It gives rise to a dramatic enhancement of the energy-transport efficiency, which may largely overcome 100%. This effect, which improves the understanding of transport phenomena in experimentally relevant complexes, suggests a particularly promising mechanism for quantum energy management.