US-led forces were tonight occupying the centre of Fallujah after a day of intense street-to-street combat.
The Pentagon said at least 10 US and two Iraqi soldiers had died since the offensive began yesterday.
There has been no word on civilian casualties, though Fallujah residents said a US air strike hit a clinic in a central district, killing some medical staff and patients.
A US military spokesman estimated that 42 Iraqis were killed across the city in bombardment and skirmishes before the main assault began yesterday but there were no firm figures today.
After sunset US tanks and armoured personnel carriers in the northern part of Falluja came under fierce assault from rebels firing rocket-propelled grenades and AK-47 rifles. Residents said explosions echoed in the night, but it appeared most large-scale fighting had eased.
Thousands of US forces flooded into the city at dawn today following heavy shelling over a number of days.
Residents said the sound of bombardment and explosions eased off around 4.15 p.m. (local time) after a night and day of intense fighting between US-led forces and insurgents.
At least two American tanks were engulfed in flames, witnesses said.
Overnight the skies over Fallujah lit up with flashes of air and artillery barrages as American forces laid siege to the city that had become the major sanctuary for Islamic extremists who fought Marines to a standstill last April.
US troops, backed by tanks and Humvees, had advanced on the city slowly from two sides, the northeastern Askari neighbourhood and the Jolan neighbourhood, a warren of alleyways where Sunni militants have dug in.
Just to the north of Jolan, Iraqi troops deployed with US forces took over the strategic train station, opening the breach that allowed the Army and Marines to penetrate toward the city centre.
Artillery, tanks and warplanes had pounded the district's northern edge overnight, softening the defences and trying to set off any bombs or boobytraps planted by the militants.
As battles raged in Falluja, insurgents hit back elsewhere with bloody attacks in Baquba and other cities in which some 60 people have been killed since last night.
Interim Prime Minister Mr Iyad Allawi, making fresh use of emergency powers he activated on Sunday, imposed a night curfew on Baghdad for an indefinite period.
A statement said the curfew would be in force from 10:30 p.m. to 4 a.m. with immediate effect. The military has given no figures for US casualties since the assault on Falluja began yesterday.