Clashes in the city, considered the main stronghold of the insurgency in Iraq, had died down by Tuesday evening, an AFP correspondent in Fallujah said, barely 24 hours after the massive US-Iraqi operation was launched to retake the city from the rebels.
Earlier during the day, Iraq's official Sunni political party quit the government in protest over the assault on Fallujah.
Underlining the chronic instability prevailing in Iraq's heartland, Prime Minister Iyad Allawi's office also announced a night-time curfew in Baghdad.
In Fallujah, only occasional tracer fire and the sound of crying babies and hysterical laughter played eerily by US psychological operations forces could be heard.
The battle to reclaim the rebel enclave spread out through neighbourhoods and alleyways from the north towards the centre as marines knocked down walls, barged into houses or crouched outside, meeting minimal resistance.
"The military controls one-third of the city," a high-ranking marine officer told AFP earlier in the day.
An exact casualty toll was unavailable from the city, where estimates for the number of its 300,000-strong population who fled ahead of the long-threatened assault vary widely from 20 percent to 90 percent.
US Lieutenant General Thomas Metz said "about a dozen" US troops had been killed so far in the offensive, without giving a precise toll. "I would not want to characterise it beyond that. It is light," he said.
The marines said more than 20 rebels were killed in a day of fighting, half of them by sniper fire, in their section of the enclave in the north-west.
Some bodies were buried under buildings that had been floored by artillery and ground bombardments, the AFP reporter said.
"As for casualties on the insurgents' side I can tell you that they are dying. A lot of them are dying and this is a good thing," Marine Spokesman Lieutenant Lyle Gilbert said.
"We are downing them," said Major Todd Desgrosseilliers, an executive officer with the Marines. "We are using good old American firepower."
But several countries urged the United States to exercise caution to avoid civilian casualties, with Russia saying that the action must not lead to the "suffering of the Iraqi people".
The extremist army in Iraq also ordered militants to attack some 20 targets in Iraq in reprisal for the Fallujah offensive, a statement published on its Web site said.
In a two-pronged assault on Fallujah that began late Monday, thousands of US troops, followed by Iraqi soldiers, poured into the north-western Jolan neighbourhood and the Askari district in the north-east. Fearful of roadside bombs - a favoured weapon of the insurgents - as they entered into Jolan, the Marines smashed through a railway line and ploughed through fields to avoid using the main roads.
They moved house-to-house through the neighbourhood, seen as the heart of rebel activity in Fallujah, spraying rounds of machine gun fire at buildings from where militants fought back with small arms fire.
Despite being a residential district, Jolan was a wasteland of shattered glass and rubble, with smoke filling the horizon. Not a civilian was in sight. Chickens and roosters ran around amid a constant clatter of Kalashnikov fire and mortar rounds.
Fighters, their faces covered in scarves, were firing gunshots from run-down houses at US soldiers before rushing on to escape a punishing barrage of American artillery and missiles, an AFP correspondent in the city said.
A smattering of specially trained Iraqi forces accompanied the Marines, while many more were poised on the outskirts of the city, preparing to enter.
Sunni and Shia figures have condemned the assault, dubbed Operation Dawn, with Iraq's main Sunni political party quitting the government in protest at the offensive, which is aimed at bringing greater security ahead of elections planned for January.
Earlier, Iraq's official Sunni Muslim political party quit the US-backed government on Tuesday in protest over the assault on Falluja, but its only member to hold a ministerial post abandoned the group to keep his government position.
"The party made the decision to withdraw from the government because of the military offensive in Falluja, but I don't share this opinion and decided to quit the party and remain in my post," Minister of Industry Hashem Al-Hassani said.
"To withdraw would not serve the interests of the Iraqi people."
US-led forces have moved deep into Falluja since launching a major assault on Monday to retake the rebel bastion as the US-backed government struggles to restore order ahead of national elections planned for January.
A spokesman for the Islamic party, Iyad Samarrai, said top party officials had met earlier in the day with Prime Minister Iyad Allawi to demand stoppage to the Falluja offensive and try to negotiate a peaceful settlement with insurgents there.
Some 20,000 US and Iraqi troops have been massing around Fallujah since mid-October and the offensive finally erupted a day after the government declared a state of emergency across most of Iraq.
US commanders estimate that 2,000 to 2,500 fighters, some loyal to Iraq's most wanted man, Abu Musab al-Zarqawi, are inside the city and its surrounding areas, ready to fight.
The military believes that another 10,000 men could join in the battle.
In Ramadi, just west of Fallujah, armed men massed on the main street after US snipers withdrew from their positions and banners were hung in the windows proclaiming solidarity with the insurgents in the neighbouring city.
"The residents of Ramadi condemn the attack against Fallujah and we appeal to the inhabitants of Ramadi to wage Jihad against the American occupants, who want to eradicate Islam," one man who did not want to be named said.
Ramadi was the scene of clashes between US forces and insurgents during the last 24 hours. At least seven Iraqis were killed, medical sources said. Mosul was also the scene of unrest, with two US military personnel killed when their base in the northern Iraqi city came under mortar fire.
Meanwhile, four attackers were killed and 14 people wounded in an ambush on two Iraqi police stations on Tuesday, a day after at least 13 people died when a Baghdad hospital was car bombed, officials said.