Sleep quality is related to people’s physical and mental health, so an accurate assessment of sleep quality is key to recognizing sleep disorders and taking effective interventions. To address the shortcomings of traditional manual and automatic staging methods, such as being time-consuming and having low classification accuracy, an automatic sleep staging method based on the null space pursuit (NSP) decomposition algorithm of single-channel electroencephalographic (EEG) signals is proposed, which provides a new way for EEG signal decomposition and automatic identification of sleep stages. First, the single-channel EEG signal data from the Sleep-EDF database, DREAMS Subject database, and Sleep Heart Health Study database (SHHS), available on PhysioNet, were preprocessed, respectively. Second, the preprocessed single-channel EEG signals were decomposed by the NSP algorithm. Third, we extracted nine features in the time domain of the nonlinear dynamics and statistics from the original EEG signal and the six simple signals that were decomposed. Finally, the extreme gradient boosting (XGBOOST) algorithm was used to construct a classification model to classify and identify the 63 extracted EEG signal features for automatic sleep staging. The experimental results showed that, on the Sleep-EDF database, the accuracy of four and five categories were 93.59% and 92.89%, respectively; on the DREAMS Subject database, the accuracy rates of four and five categories were 91.32% and 90.01%, respectively; on the SHHS database, the accuracy rates of four and five categories were 90.25% and 88.37%, respectively. The experimental results show that the automatic sleep staging model proposed in this work has high classification accuracy and efficiency, as well as strong applicability and robustness.