Lt. Col. Steve Stover, a U.S. military spokesman, says the clashes broke out after a U.S. patrol was attacked about 9:30 a.m. with rocket-propelled grenades and machine guns.
He says six U.S. soldiers were wounded in Tuesday's fighting but none of their injuries were life-threatening.
The violence appeared to be a continuation of the weekend's heavy clashes involving attack helicopters and Abrams main battle tanks in which about 45 militants and four U.S. soldiers have died.
A showdown between the Iraqi government and the Mahdi Army militia - led by anti-American cleric Muqtada al-Sadr - has increasingly drawn U.S. forces into battle. American commanders are particularly focused on trying to curb a rise in mortar and rocket attacks on the heavily fortified Green Zone in central Baghdad that houses Iraq's government and many foreign embassies.
Overnight clashes resulted in 42 injuries, officials at the Imam Ali and al-Sadr general hospitals said. Eight more were killed and 25 wounded in continuing firefights on Tuesday morning, said the officials who spoke on condition of anonymity because they were not authorized to speak to the media.
AP Television News footage showed men helping women cross the street and children running for cover behind blast walls amid gunshots.
Men helped carry several blood-soaked injured people onto stretchers to a local emergency hospital. Outside the hospital, the dead were placed inside plain wooden coffins.
Also in Baghdad, a senior government official was killed in a roadside bombing in the north of the city.
Dhia Jodi Jaber, director general at the Ministry of Labor and Social Affairs, was hit by a roadside bomb as he left his home on Tuesday morning, the ministry's spokesman Abdullah al-Lami said.
Insurgents frequently target governmental officials and institutions in a bid to disrupt the government's work.
In the southern city of Basra, where the government began its crackdown on Shiite militias on March 25, Iraqi military commander Lt. Gen. Mohan al-Fireji announced the discovery of a huge weapons cache containing roadside bombs, mortar launchers and Iranian-made weapons.
More details on the amount of weapons or how authorities knew they were Iranian-made were not immediately available.
In other developments: