Motivated by the advances in deep learning techniques, the application of Unmanned Aerial Vehicle (UAV)-based object detection has proliferated across a range of fields, including vehicle counting, fire detection, and city monitoring. While most existing research studies only a subset of the challenges inherent to UAV-based object detection, there are few studies that balance various aspects to design a practical system for energy consumption reduction. In response, we present the E$^3$-UAV, an edge-based energy-efficient object detection system for UAVs. The system is designed to dynamically support various UAV devices, edge devices, and detection algorithms, with the aim of minimizing energy consumption by deciding the most energy-efficient flight parameters (including flight altitude, flight speed, detection algorithm, and sampling rate) required to fulfill the detection requirements of the task. We first present an effective evaluation metric for actual tasks and construct a transparent energy consumption model based on hundreds of actual flight data to formalize the relationship between energy consumption and flight parameters. Then we present a lightweight energy-efficient priority decision algorithm based on a large quantity of actual flight data to assist the system in deciding flight parameters. Finally, we evaluate the performance of the system, and our experimental results demonstrate that it can significantly decrease energy consumption in real-world scenarios. Additionally, we provide four insights that can assist researchers and engineers in their efforts to study UAV-based object detection further.