"It is believed that the suspected terrorists have close ties to Iranian terror networks and are responsible for facilitating the flow of lethal aid into Iraq."
The statement said US soldiers faced "significant" resistance during the pre-dawn raids, including roadside bomb attacks as well as fire from rocket-propelled grenades and light weapons.
There were no US casualties, it said.
Residents said US attack helicopters fired missiles at targets in the densely-populated area of two million people in northeastern Baghdad.
Sadr City is the stronghold of Moqtada al-Sadr, a Shia leader who wants US forces to leave Iraq.
Washington blames rogue elements of Sadr's powerful Mehdi Army militia of using advanced roadside bombs, made from components smuggled in from Iran, to target US troops.