The operation began at dawn and lasted several hours
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Iraqi forces backed by US helicopter gunships have fought insurgents in the Sunni-dominated Haifa Street area of the capital, Baghdad.
The US said it was part of the new plan to restore security to the capital.
An Iraqi official said up to 30 militants were killed and 27 detained. One US soldier was killed.
The battle came hours after President George W Bush urged the US to give his new strategy a chance to work, warning of "grievous consequences" of failure.
In his annual State of the Union address to Congress, Mr Bush said that if the US stepped back before Baghdad was secure, "the Iraqi government would be overrun by extremists from all sides".
The new US strategy is to take control of insurgent strongholds and then to hold them, rather than withdrawing and allowing the areas to be re-taken by anti-US insurgents and sectarian militias.
An advance guard of 3,200 troops arrived in Baghdad on Sunday, part of President Bush's plan to send an extra 21,500 troops to Iraq.
Machine-gun fire
The BBC's Mike Wooldridge in Baghdad says the latest operation began at dawn and for several hours afterwards loud explosions and frequent bursts of heavy machine-gun fire could be heard from the Haifa Street district.
Witnesses said insurgents used rocket-propelled grenades and mortars and machine-guns against a joint force of US and Iraqi troops.
Haifa Street is a major Baghdad thoroughfare lined with high-rise buildings.
"A lot has been coming from high-rise buildings," US military spokesman Maj Steven Lamb told Reuters news agency. "We are firing at terrorists in those buildings."
A statement from the US military said that one soldier was killed and two others injured in a combined security operation in Baghdad.
There were unconfirmed reports of civilian casualties.
Earlier, gunfire and mortar rounds were heard coming from another district, al-Fadl, where five US civilians died on Tuesday when a US helicopter crashed after apparently coming under fire.
Elsewhere in the city, gunmen killed a well-known professor and economist who presented a programme on Sharqiya television.
Diya al-Meqoter was shot several times in the head and chest
He was known for supporting poor people needing loans to set up businesses, and he also headed Iraq's consumer association, a non-government agency which campaigned for fair pricing.
