Environment-assisted quantum transport (ENAQT) is the possibility of an external environment to enhance transport efficiency of quantum particles. This idea has generated much excitement over recent years, especially due to the experimentally motivated possibility of ENAQT in photosynthetic exciton transfer complexes. Many theoretical calculations have shown ENAQT, but the explanations for its origin differ, and a universal explanation has been elusive. Here we demonstrate a universal origin for ENAQT in quantum networks with a dephasing environment, based on a relation between exciton current and occupation within a Markovian open quantum system approach. We show that ENAQT appears due to two competing processes, namely, the tendency of dephasing to make the exciton population uniform, and the formation of an exciton density gradient, defined by the source and the sink. Furthermore, we find a geometric condition on the network for the appearance of ENAQT, relevant to natural and artificial systems.