Sadr City, a Shiite slum, is one of the strongholds of Muqtada al Sadr, the firebrand cleric who slammed the Iraqi interim government.
The 10 soldiers were members of Task Force Baghdad, which is made up primarily of the Texas-based 1st Cavalry Division.
Reports from the scene described the wreckage of a Humvee engulfed in flames and smoke as U.S. troops cordoned off the area and evacuated the wounded.
As soldiers tried to drag victims from the burning wreckage, gunmen opened fire, reports CBS News Correspondent Charles D'Agata.
Also Friday, the U.S. military and Shiite militia loyal to a radical cleric agreed Friday to withdraw from areas around holy shrines south of Baghdad and turn over security to Iraqi police, an Iraqi official said.
In a televised address to the nation, the new prime minister, Iyad Allawi, said that a rapid U.S. withdrawal from Iraq after the handover of sovereignty would be a "major disaster" since Iraqis are not ready to handle their own security.
The reported agreement with cleric Muqtada Al-Sadr in the twin cities of Najaf and Kufa is broadly similar to the accord that ended the bloody, three-week Marine siege of Fallujah, a Sunni insurgent stronghold west of Baghdad. The Marines struck a deal there to lift the siege and hand over security to an Iraqi force commanded by former officers from Saddam Hussein's army.
In other developments:
Local authorities in Najaf and Kufa hope the presence of more Iraqi police will defuse tensions and allow the agreement to take hold where an earlier deal with al-Sadr did not. Many Iraqi police deserted when al-Sadr launched his uprising two months ago, handing the streets over to the cleric's al-Mahdi Army.
Najaf Gov. Adnan al-Zurufi announced Friday's agreement, in which he said both sides agreed to withdraw from around the shrines in Najaf and Kufa. The Americans refuse to negotiate directly with al-Sadr and said they agreed to a request by al-Zurufi to reposition forces in the interest of peace.
But by late Friday, militiamen showed no intention of leaving the Imam Ali Shrine in Najaf. They told The Associated Press that they were instructed only to put their weapons up — not to withdraw.
About 15 Iraqi police moved a half a mile from the shrine, but militia officials said they would not withdraw from the area without firm guarantees that American troops would not hunt them down.
The agreement would be a major step toward ending a two-month Shiite uprising in the south and parts of Baghdad as the U.S.-led occupation draws to a close and a new Iraqi government prepares to take power June 30.
Al-Sadr's rebellion began after the U.S.-led occupation authority closed his newspaper, arrested a key aide and announced a warrant for his arrest in the April 2003 murder of a moderate cleric in Najaf.
On May 27, Shiite leaders announced al-Sadr had agreed to remove his fighters from the streets and send home any militiamen who lived outside the Najaf and Kufa areas if the Americans pulled back too.
That announcement failed to stop the daily clashes between U.S. soldiers and militiamen, especially in Kufa. U.S. officials accused the militia of firing mortar shells at the U.S. base between the two cities.
Friday's deal was announced as the U.N. Security Council debates a U.S.-British resolution outlining a blueprint for the transfer of power. Key council members want the Iraqis to have the final say in offensive military operations by U.S. and international troops who will remain after June 30.
The United States and Britain revised their resolution Friday, giving Iraq's interim government authority to order the U.S.-led multinational force to leave at any time.
Col. Brian May, commander of the 2nd Armored Cavalry Regiment, said U.S. commanders agreed to stay away from "sensitive areas" — the Imam Ali Shrine in Najaf and the Kufa mosque where al-Sadr preaches — because al-Zurufi assured them the militia had "been reduced to the point where the legitimate Iraqi security forces can move in to those very sensitive areas."
"It's an Iraqi solution to the problem," he said.
Despite the assurances, militiamen still manned the gates to the shrine and searched visitors well after an afternoon deadline passed for their departure.
Men in yellow shirts and badges identifying them as members of al-Sadr's militia dispersed crowds and urged people to go away. A banner outside al-Sadr's office read: "Al-Sadr doesn't compromise" and "We all resist."
Al-Sadr failed to mention the deal in a statement read on his behalf in the Kufa mosque. The statement denounced the interim Iraqi government and insisted on an elected leadership for Iraq.
"America has taken upon itself to appoint a prime minister and a president of the nation under the cover of the United Nations," al-Sadr's message said. "It has done that with impertinence and domination. The government must be elected, and I will never accept anything beneath that."
He said he could not imagine "any reasonable person would ever accept" a government "which comes from no less than the occupying power."
Prime Minister Allawi told listeners that Iraq would never accept occupation and looked forward to having the U.N. Security Council adopt "a new resolution regarding the transfer of full sovereignty to the interim Iraqi government."
Allawi, a former exile leader who had close ties to the CIA and State Department, also said that security was a paramount challenge facing the new government and that it would work toward national unity after the divisions created by the war, tyranny and military occupation.
He pledged former Baathists could live with dignity if they had not committed any crimes.
Allawi, a Shiite Muslim, also expressed appreciation to the country's most influential Shiite cleric, Grand Ayatollah Ali al-Husseini al-Sistani, who has given a tacit endorsement of the new government.
He also said the newly appointed president, Ghazi al-Yawer, a Sunni, would attend the upcoming Group of Eight meeting to improve the country's economy.