A new method of power grid attack, the False Data Injection Attack (FDIA), seriously threatens the safe operation of smart grids. The explosive growth of data has brought huge challenges to centralized FDIA detection methods. This paper proposes a detection method based on edge computing, which divides the system into multiple subsystems, and sets edge node detectors in the subsystems for data collection and detection. Combined with deep learning methods, a CNN-LSTM detecting model is constructed to extract the characteristics of the data, and the training process of the model is placed on the central node to achieve efficient and low-latency FDIA detection. Finally, the proposed edge detection method is verified in the IEEE 14-node and IEEE 39-node test systems for different attack intensities. Compared with the centralized detection method, the results show that the advanced edge detection method can achieve a significant drop in detection time and memory consumption. This work is supported by the National Natural Science Foundation of China (No. 41671431).