The striking efficiency of energy transfer in natural photosynthetic systems and the recent evidence of long-lived quantum coherence in biological light harvesting complexes has triggered much excitement, due to the evocative possibility that these systems - essential to practically all life on earth -- use quantum mechanical effects to achieve optimal functionality. A large body of theoretical work has addressed the role of local environments in determining the transport properties of excitons in photosynthetic networks and the survival of quantum coherence in a classical environment. Nonetheless, understanding the connection between quantum coherence, exciton network geometry and energy transfer efficiency remains a challenge. Here we address this connection from the perspective of heat transfer within the exciton network. Using a non-equilibrium open quantum system approach and focusing on the Fenna-Matthews-Olson complex, we demonstrate that finite local dephasing can be beneficial to the overall power output. The mechanism for this enhancement of power output is identified as a gentle balance between quantum and classical contributions to the local heat flow, such that the total heat flow is directed along the shortest paths and dissipation is minimized. Strongly related to the spatial network structure of the exciton transfer complex, this mechanism elucidates how energy flows in photosyntetic excitonic complexes.