Mobile edge computing (MEC) has currently recognized as capable computing model to provide a delay-sensitive, computation-intensive services for mobile users. The existing methods suffered from high delay and energy consumption (EC). Therefore, the logistic map-based gazelle optimization algorithm (LMGOA) is proposed to reduce EC by optimizing the use of a task offloading solution and resource management in a highly dynamic environment. The logistic map used for initialization is incorporated to reinforce the disorder and equality of the initial solution across the space, cause a superior optimization in search space. This helps in preventing the algorithm from converging earlier to a solution and enhance the algorithms capability to find the best solutions for energy-efficient MEC. The delay, EC and throughput are taken to estimate the LMGOA performance. The LMGOA achieved delay of 32.65 s, EC of 16.509 J and throughput of 7692.3 Mbps for 1000 no. of mobile devices which is better than existing approaches.