Objective: The aim of the study is to develop a novel method for improved diagnosis of obstructive sleep apnea-hypopnea syndrome (OSAHS) in clinical or home settings, with the focus on achieving diagnostic performance comparable to the gold-standard polysomnography (PSG) with significantly reduced monitoring burden. Methods: We propose a method using millimeter-wave radar and pulse oximeter for OSAHS diagnosis (ROSA). It contains a sleep apnea-hypopnea events (SAE) detection network, which directly predicts the temporal localization of SAE, and a sleep staging network, which predicts the sleep stages throughout the night, based on radar signals. It also fuses oxygen saturation (SpO2) information from the pulse oximeter to adjust the score of SAE detected by radar. Results: Experimental results on a real-world dataset (>800 hours of overnight recordings, 100 subjects) demonstrated high agreement (ICC=0.9870) on apnea-hypopnea index (AHI) between ROSA and PSG. ROSA also exhibited excellent diagnostic performance, exceeding 90% in accuracy across AHI diagnostic thresholds of 5, 15 and 30 events/h. Conclusion: ROSA improves diagnostic accuracy by fusing millimeter-wave radar and pulse oximeter data. It provides a reliable and low-burden solution for OSAHS diagnosis. Significance: ROSA addresses the limitations of high complexity and monitoring burden associated with traditional PSG. The high accuracy and low burden of ROSA show its potential to improve the accessibility of OSAHS diagnosis among population.