This paper proposes a novel offloading and super-resolution (SR) control scheme for energy-efficient mobile augmented reality (MAR) in multi-access edge computing (MEC) using SR as a promising generative artificial intelligence (GAI) technology. Specifically, SR can enhance low-resolution images into high-resolution versions using GAI technologies. This capability is particularly advantageous in MAR by lowering the bitrate required for network transmission. However, this SR process requires considerable computational resources and can introduce latency, potentially overloading the MEC server if there are numerous offload requests for MAR services. In this context, we conduct an empirical study to verify that the computational latency of SR increases with the upscaling level. Therefore, we demonstrate a trade-off between computational latency and improved service satisfaction when upscaling images for object detection, as it enhances the detection accuracy. From this perspective, determining whether to apply SR for MAR, while jointly controlling offloading decisions, is challenging. Consequently, to design energy-efficient MAR, we rigorously formulate analytical models for the energy consumption of a MAR device, the overall latency and the MAR satisfaction of service quality from the enforcement of the service accuracy, taking into account the SR process at the MEC server. Finally, we develop a theoretical framework that optimizes the computation offloading and SR control problem for MAR clients by jointly optimizing the offloading and SR decisions, considering their trade-off in MAR with MEC. Finally, the performance evaluation indicates that our proposed framework effectively supports MAR services by efficiently managing offloading and SR decisions, balancing trade-offs between energy consumption, latency, and service satisfaction compared to benchmarks.