In recent years, the huge expansion of Datacenters (DC) to execute billions of end-user applications in real-time leads to a large amount of energy consumption across the globe. So, the traditional TCP/IP-based networks which are being used for DC inter-connections are facing challenges of managing stringent Quality-of-Service (QoS) requirements of different applications of the end-users and service providers. Moreover, the existing solutions rely on distributed architecture and do not scale for large scale data centers. The issue of high power consumption at DC arises with the increase in the number of nodes and links in the network. Also, it becomes problematic on the DC whenever the underlying network resources (switches and routers) are not efficiently utilized at the time of peak data traffic resulting in high operational cost of energy utilization. However, Software-Defined Networking (SDN) emerges as one of the leading technologies to address the aforementioned issues using the programmable switches and controllers. Inspired from these facts, in this paper, we have formulated the Energy-Aware Routing (EAR) problem of DCs as a Mixed Integer Non-Linear Programming (MINLP) for which an Energy-Efficient Fast Flow Forwarding (EnFlow) scheme is designed. The EnFlow scheme uses the power-saving mode of the network to solve the EAR problem. It has three modules namely- priority scheduling, routing, and re-routing. The first module works according to the First-in-First-Out Push Out Priority (FIFO-POP) scheduling using the multiple OpenFlow switches. The FIFO-POP is designed to save the energy usage of multiple switches by reducing the average waiting time of incoming packets in the queue buffers. The second module is based upon an efficient flow re-routing for a new node and link adaptation to provide the maximum bandwidth to the wired links. The third module is based upon the meta-heuristic Ant Colony Routing (ACR) to execute the stochastic decision policy on the network controller for computation of the shortest path of the forwarding nodes. The proposed EnFlow scheme is simulated using the data traces of 34 cities of NorthAmerica zone with Omnet++ 5.1 using various performance evaluation metrics. The results obtained demonstrated that the proposed EnFlow scheme is 24.55% and 71.15% more energy-efficient in comparison to the RE-FPR and ILP-EAR schemes. Also, it consumes 9.72% and 40.83% lower energy in comparison to the FFHA and EXR schemes respectively.