State estimation is of considerable significance for the power system operation and control. However, well-designed false data injection attacks can utilize blind spots in conventional residual-based bad data detection methods to manipulate measurements in a coordinated manner and thus affect the secure operation and economic dispatch of grids. In this paper, we propose a detection approach based on an autoencoder neural network. By training the network on the dependencies intrinsic in normal operation data, it effectively overcomes the challenge of unbalanced training data that is inherent in power system attack detection. To evaluate the detection performance of the proposed mechanism, we conduct a series of experiments on the IEEE 118-bus power system. The experiments demonstrate that the proposed autoencoder detector displays robust detection performance under a variety of attack scenarios.