Mobile edge computing (MEC) is considered a promising technique that prolongs battery life and enhances the computation capacity of mobile devices (MDs) by offloading computation-intensive tasks to the resource-rich cloud located at the edges of mobile networks. In this study, the problem of energy-efficient computation offloading with guaranteed performance in multiuser MEC systems was investigated. Given that MDs typically seek lower energy consumption and improve the performance of computing tasks, we provide an energy-efficient computation offloading and transmit power allocation scheme that reduces energy consumption and completion time. We formulate the energy efficiency cost minimization problem, which satisfies the completion time deadline constraint of MDs in an MEC system. In addition, the corresponding Karush–Kuhn–Tucker conditions are applied to solve the optimization problem, and a new algorithm comprising the computation offloading policy and transmission power allocation is presented. Numerical results demonstrate that our proposed scheme, with the optimal computation offloading policy and adapted transmission power for MDs, outperforms local computing and full offloading methods in terms of energy consumption and completion delay. Consequently, our proposed system could help overcome the restrictions on computation resources and battery life of mobile devices to meet the requirements of new applications.