Deep neural network (DNN) models are increasingly popular in edge video analytic applications. However, the compute-intensive nature of DNN models pose challenges for energy-efficient inference on resource-constrained edge devices. Most existing solutions focus on optimizing DNN inference latency and accuracy, often overlooking energy efficiency. They also fail to account for the varying complexity of video frames, leading to sub-optimal performance in edge video analytics. In this paper, we propose an Energy-Efficient Early-Exit (E4) framework that enhances DNN inference efficiency for edge video analytics by integrating a novel early-exit mechanism with dynamic voltage and frequency scaling (DVFS) governors. It employs an attention-based cascade module to analyze video frame diversity and automatically determine optimal DNN exit points. Additionally, E4 features a just-in-time (JIT) profiler that uses coordinate descent search to co-optimize CPU and GPU clock frequencies for each layer before the DNN exit points. Extensive evaluations demonstrate that E4 outperforms current state-of-the-art methods, achieving up to 2.8x speedup and 26% average energy saving while maintaining high accuracy.