The speed of the U.S.-led offensive may indicate that most Sunni militants had already left the city to reposition themselves for fighting elsewhere, officers said.
"Many armed groups" in the city had asked to surrender and Iraqi authorities "will extend amnesty" to those who have not committed major crimes, said government spokesman Thair al-Naqeeb, according to the Associated Press.
Kidnappers, meanwhile, snatched three members of interim Prime Minister Iyad Allawi's family.
At least 71 militants have been killed during the three days of the offensive, the military said. As of Tuesday night, 10 U.S. troops and two members of the Iraqi security forces had been killed, the AP reported. Twenty-five American troops and 16 Iraqi soldiers had been wounded, according to Marine reports on Wednesday.
Maj. Francis Piccoli of the 1st Marine Expeditionary Force said U.S. forces now control 70 percent of the city and had pushed insurgents to a narrow section along the main east-west highway bisecting the city, reported the AP.
Fighting in other parts of Iraq continued, resulting in the deaths of at least 18 people, including a U.S. soldier and a foreign contractor. A curfew was imposed on the northern city of Mosul as U.S. and Iraqi forces clashed with gunmen there. Fighting also continued in and around the capital Baghdad.
Armed men kidnapped a 75-year-old cousin of Allawi, the cousin's wife and their daughter-in-law from their home in Baghdad, according to Reuters.
A previously unknown military group calling itself the Ansar al-Jihad threatened on its Web site to kill the hostages within 48 hours unless the Fallujah siege ended.
Allawi's office responded to the abductions by saying the government would not give in. "This is yet another criminal act by terrorists and will not thwart the determination of the government to combat terrorism," it said in a statement.
Allawi, backed by the United States and Britain, is determined to hold national assembly elections by January.
But political unrest is brewing, particularly since the massive offensive in Fallujah began. A prominent group of Sunni clerics has called on Iraqis to boycott the national elections.
"The clerics call on honorable Iraqis to boycott the upcoming election that is to be held over the bodies of the dead and the blood of the wounded in cities like Fallujah," said Harith al-Dhari, director of the Muslim Scholars Association, a group of Sunni clerics that says it represents 3,000 mosques, according to The New York Times.
Hours earlier, the group issued a religious edict ordering Iraqi security forces not to take part in the siege.
And on Tuesday the Iraqi Islamic Party withdrew from the interim government, the Times reported.
Mohsen Abdul Hameed, a Sunni interim National Assembly member, said, "After the attack on Fallujah, we decided to withdraw from the government because our presence in the government will be judged by history."
Allawi met privately with Abdul Hameed hours later, but the party stuck to its position. An aide later said that it was not clear if the group would take part in the elections.
"We haven't decided to withdraw from the elections; we're still going forward with the process," the aide, Ayad al-Samarrai, said. "But it will all depend on the general situation in Iraq."
A Sunni boycott of the elections would threaten their legitimacy.