Mobile edge computing (MEC) has shown tremendous potential as a means for computationally intensive mobile applications by partially or entirely offloading computations to a nearby server to minimize the energy consumption of user equipment (UE). However, the task of selecting an optimal set of components to offload considering the amount of data transfer as well as the latency in communication is a complex problem. In this paper, we propose a novel energy-efficient deep learning based offloading scheme (EEDOS) to train a deep learning based smart decision-making algorithm that selects an optimal set of application components based on remaining energy of UEs, energy consumption by application components, network conditions, computational load, amount of data transfer, and delays in communication. We formulate the cost function involving all aforementioned factors, obtain the cost for all possible combinations of component offloading policies, select the optimal policies over an exhaustive dataset, and train a deep learning network as an alternative for the extensive computations involved. Simulation results show that our proposed model is promising in terms of accuracy and energy consumption of UEs.