This study proposes a method of automatically classifying sleep apnea/hypopnea events based on sleep states and the severity of sleep-disordered breathing (SDB) using photoplethysmogram (PPG) and oxygen saturation (SpO2) signals acquired from a pulse oximeter. The PPG was used to classify sleep state, while the severity of SDB was estimated by detecting events of SpO2 oxygen desaturation. Furthermore, we classified sleep apnea/hypopnea events by applying different categorisations according to the severity of SDB based on a support vector machine. The classification results showed sensitivity performances and positivity predictive values of 74.2% and 87.5% for apnea, 87.5% and 63.4% for hypopnea, and 92.4% and 92.8% for apnea + hypopnea, respectively. These results represent better or comparable outcomes compared to those of previous studies. In addition, our classification method reliably detected sleep apnea/hypopnea events in all patient groups without bias in particular patient groups when our algorithm was applied to a variety of patient groups. Therefore, this method has the potential to diagnose SDB more reliably and conveniently using a pulse oximeter.