Three policemen, including two officers, and three of the arrested suspects were wounded in the gunfight that started at dawn after security forces backed by armoured vehicles stormed a house where the suspects had taken refuge.
State television later reported that one of the wounded Islamists had died of his injuries.
"Security forces ... have liquidated a group of terrorists who had taken refuge in a number of locations in Mubarak Al-Kabir district where they started shooting at security men who hunted them," the statement said.
Security sources and witnesses told AFP that policemen fought in the streets with the militants who ignored repeated calls through loudspeakers to surrender.
State television said one of those killed was a Saudi national while security sources told AFP two others were Bidoon, or stateless Arabs.
One of those arrested was Amer Khlaif al-Enezi, a former mosque preacher and the alleged spiritual leader of militants who have been involved in a series of clashes with security forces this month. His brother Nasser was killed in a gunfight with police on Sunday, the interior ministry said.