"It was these militants who initiated the engagement by attacking US soldiers," he said.
A resident of Sadr City said "the fighting was intense" and that "four houses [were] heavily damaged".
Several rockets or mortar rounds also struck the Iraqi capital's heavily fortified 'Green Zone' government compound, as the fighters took advantage of the absence of US air cover during a severe sandstorm, witnesses said.
The predominantly Shia Sadr City district of the capital is largely controlled by the al-Mahdi Army of Muqtada al-Sadr, a Shia leader.
Iraqi officials said that three women and a child were among the dead following the clashes.
"The enemy continues to show little regard for innocent civilians, as they fire their weapons from within houses, alleyways and rooftops upon our soldiers," Colonel Allen Batschelet, US military commander, said in a statement.
Salah al-Obaidi, a spokesman for Al-Sadr Movement, has accused Iran with what he dubbed sharing of power and status with the United States in Iraq.
Obaidi said that Iran has agreed behind the scenes not to object to the long-term convention between the Iraqi government and America.Iraqi and US forces have been fighting against Shia armed group since March 25 in Sadr City, as well as the southern city of Basra.
Hundreds of Shia fighters and civilians have been killed in the fighting.
At least 18 US soldiers have also been killed in Baghdad since the government led crackdown against Shia fighters was launched.