Police said the attacker stabbed the policeman at a Muscat shopping centre, adding that the assailant "was in possession of a document indicating that he was mentally ill".
One officer told Omani public television that police had gone to the shopping centre after reports of suspicious behaviour and confronted the man at a fast-food restaurant.
The suspect pulled a knife and stabbed one officer, delivering a fatal wound. Two other officers were wounded and the assailant was arrested.