Monitoring the depth of unconsciousness during anesthesia is beneficial in both clinical settings and neuroscience investigations to understand brain mechanisms. Electroencephalogram (EEG) has been used as an objective means of characterizing brain altered arousal and/or cognition states induced by anesthetics in real-time. Different general anesthetics affect cerebral electrical activities in different ways. However, the performance of conventional machine learning models on EEG data is unsatisfactory due to the low Signal to Noise Ratio (SNR) in the EEG signals, especially in the office-based anesthesia EEG setting. Deep learning models have been used widely in the field of Brain Computer Interface (BCI) to perform classification and pattern recognition tasks due to their capability of good generalization and handling noises. Compared to other BCI applications, where deep learning has demonstrated encouraging results, the deep learning approach for classifying different brain consciousness states under anesthesia has been much less investigated. In this paper, we propose a new framework based on meta-learning using deep neural networks, named Anes-MetaNet, to classify brain states under anesthetics. The Anes-MetaNet is composed of Convolutional Neural Networks (CNN) to extract power spectrum features, and a time consequence model based on Long Short-Term Memory (LSTM) networks to capture the temporal dependencies, and a meta-learning framework to handle large cross-subject variability. We use a multi-stage training paradigm to improve the performance, which is justified by visualizing the high-level feature mapping. Experiments on the office-based anesthesia EEG dataset demonstrate the effectiveness of our proposed Anes-MetaNet by comparison of existing methods.