"We have absolutely won the fighting that started in Mogadishu this morning. We now control the whole city after we seized the last territory from warlord Qeydiid," said Sheikh Mukhtar Robow, the deputy secretary of defence for the Islamic courts.
Fighters allied to the Supreme Islamic Council of Somalia ousted their rivals loyal to warlord Abdi Hassan Awale Qeydiid, who with warlord Hussein Aidid had refused to surrender and hand over their weapons to the Islamists.
Doctors, witnesses and militia said at least 21 people were killed, including civilians, in artillery duels in southern Mogadishu, while dozens were wounded and taken to the capital's Medina and Banadir hospitals.
Witnesses said warlords' fighters fled from their positions as Islamic militants on pickup trucks equipped with machine guns established their bases, marking the end of the warlords' rule in the capital.
"Since the Islamists have a policy of not revealing their battlefield losses, we believe the overall toll could be as high as 43," said an Islamic militiaman.
Mr Aidid, also deputy prime minister in the transitional administration, was in the seat of government in Baidoa, about 250 km north-west of the capital while Mr Qeydiid fled the capital.
The Islamists, who routed the US-backed warlords from the capital on June 5, vowed to rid the country of warlords and other faction chiefs who have ruled since the government of dictator Mohamed Siad Barre was toppled in 1991.
The Islamists reject charges that they are linked with Osama bin Laden's al-Qaeda network and that they harbour foreign fighters.