Limited battery and computing resources of mobile devices (MDs) induce performance limitations in mobile edge computing (MEC) networks. Computational offloading has the capability to provide computing and storage resources to MDs for resource-intensive tasks execution. Therefore, to minimize energy consumption and service delay, MDs offload the resource-intensive tasks to nearby mobile edge server (MES) for execution . However, due to time varying network conditions and limited computing resources at MES also, the offloading decision taken by MDs may not achieve the lowest cost. In this paper, we propose an energy efficient and faster deep learning based offloading technique (EFDOT) to minimize the overall cost of MDs. We formulate a cost function which considers the energy consumption and service delay of MDs, radio resources, energy consumption and delay due to task partitioning, and computing resources of the MDs and MESs. Due to high computational overhead of this comprehensive cost function, we generate a training dataset to train a deep neural network (DNN) in order to make the decision making process faster. The proposed work finds the optimal number of components, task partitioning, and fine-grained offloading policy simultaneously. We formulate the fine-grained offloading decision problem in MEC as multi-label classification problem and propose EFDOT to minimize the computation and offloading overhead. The simulation results show high accuracy of the DNN and high performance of EFDOT in terms of energy consumption, service delay, and battery life.