"The military controls one third of the city," the officer told AFP, confirming that it was the northern part believed to be a stronghold for the insurgents in the city.
Thousands of US troops, followed by crack Iraqi soldiers, poured into the northwestern Jolan neighbourhood and the Askari district in the northeast in a two-pronged assault that started late Monday.
The action, the biggest military operation since the US-led war in March last year, came after Iraqi Prime Minister Iyad Allawi gave the nod for a full-scale offensive.
The troops faced resistance at the beginning but it soon faded, another senior military official said.
They seized Fallujah's train station in the northeast and carried on pushing towards the centre, an AFP correspondent embedded with the military said.