Recently Unmanned Aerial Vehicles (UAVs) have evolved considerably towards real world applications, going beyond entertaining activities and use. With the advent of Fifth Generation (5G) cellular networks and the number of UAVs to be increased significantly, it is created the opportunity for UAVs to participate in the realisation of 5G opportunistic networks by carrying 5G Base-Stations to under-served areas, allowing the provision of bandwidth demanding services, such as Ultra High Definition (UHD) video streaming, as well as other multimedia services. Among the various improvements that will drive this evolution of UAVs, energy efficiency is considered of primary importance since will prolong the flight time and will extend the mission territory. Although this problem has been studied in the literature as an offline resource optimisation problem, the diverse conditions of a real UAV flight does not allow any of the existing offline optimisation models to be applied in real flight conditions. To this end, this paper discusses the amalgamation of UAVs and 5G cellular networks as an auspicious solution for realising energy efficiency of UAVs by offloading at the edge of the network the Flight Control System (FCS), which will allow the optimisation of the UAV energy resources by processing in real time the flight data that have been collected by onboard sensors. By exploiting the Multi-access Edge Computing (MEC) architectural feature of 5G as a technology enabler for realising this offloading, the paper presents a proof-of-concept implementation of such a 5G-enabled UAV with softwarized FCS component at the edge of the 5G network (i.e., the MEC), allowing by this way the autonomous flight of the UAV over the 5G network by following control commands mandated by the FCS that has been deployed at the MEC. This proof-of-concept 5G-enabled UAV can support the execution of real-time resource optimisation techniques, a step-forward from the currently offline-ones, enabling in the future the execution of energy-efficient and advanced missions.