Wireless mesh networks are a special class of wireless networks that are implemented as a collection of radio nodes in a mesh pattern or topology. Unlike MANETs, the mobility of nodes is very less in the topology. Quality of service is an essential metric in the performance of mesh networks which are attributed to several parameters including optimal routing through shortest path, ability for other nodes to communicate even if a particular node in the mesh fails, minimization of packet loss and time delay, computational complexity and cost, energy. This research paper is focused toward minimization of energy taken as the objective function and a five-stage neural network is used and trained after optimizing with a genetic algorithm. The experiments have been conducted in NS2 and Qualnet environment with a varying number of mesh routers and energy computed. The performance of energy savings has been compared against conventional routing techniques like AODV and a bee colony optimization technique presented in the literature. An energy savings of 51% have been reported in the paper justifying the superiority of the hybrid G-ANN algorithm.