The energy efficiency of wired networks has received considerable attention over the past decade due to its economic and environmental impacts. However, because of the vertical integration of the control and data planes in conventional networks, optimizing energy consumption in such networks is challenging. Software-defined networking (SDN) is an emerging networking paradigm that decouples the control plane from the data plane and introduces network programmability for the development of network applications. In this work, we propose an energy-aware integral flow-routing solution to improve the energy efficiency of the SDN routing application. We consider the discreteness of link rates and pose the routing problem as a mixed-integer linear programming (MILP) problem, which is known to be NP-complete. The proposed solution is a heuristic implementation of the Benders decomposition method that routes additional single and multiple flows without resolving the routing problem. Performance evaluations demonstrate that the proposed solution achieves a close-to-optimal performance (within 3.27% error) compared to CPLEX on various topologies with less than 0.056% of CPLEX average computation time. Furthermore, our solution outperforms the shortest path algorithm by 24.12% to 54.35% in power savings.