5G networks promise to deliver an unprecedented performance that can accommodate novel services with stringent Quality of Service (QoS) requirements that were not possible with previous generations of networks. Edge Computing plays a fundamental role by providing computing resources closer to the user, reducing round trip times. However, the deployment of edge computing poses new challenges, including the energy footprint of a potentially large number of servers. Even in idle state, these servers consume a significant amount of energy, which is worth considering for reducing their energy footprint. In cloud computing environments, server shutdown during low-demand periods is a typical energy-saving strategy. However, this approach has received less attention in edge computing due to the strict latency requirements of its use cases. This work presents ODESA, an edge server shutdown strategy with polynomial time complexity that provides a trade-off between the idle energy consumption of the edge servers and energy consumed by the backhaul to route requests to active servers. Our numerical investigation shows that thanks to the reduction in idle energy consumption, ODESA reduces the total consumption by 42% over the common always-on approach during low-demand periods and 11% over 24 hours, all while meeting the latency requirements of the applications