US marine Sergeant David Sutherland, speaking to BBC News from the square in front of the Palestine hotel, said that gunfire could be heard on a half hourly basis coming from the northern part of the city.
Central Command has emphasised that some Iraqis may still be prepared to fight to stop the US advance.
There has been no sign of the officials who have until now dealt with the media on a daily basis.
Our correspondent says that all those in authority appear to have fled, leaving a potentially dangerous power vacuum.
The day began with cheering Iraqis greeting marines arriving in the Shia stronghold of Saddam City.
Already, United Nations offices have been ransacked and the Olympic Committee building - headquarters of Saddam Hussein's elder son Uday - set on fire.
Television pictures showed people kicking images of Saddam Hussein and carrying off everything from elaborate vases to office furniture.
The response in the Palestine hotel area was initially more muted, but correspondents say
US marines were later mobbed by crowds offering flowers.
Our correspondent says people would not be behaving in this way unless they were sure Saddam Hussein's grip on the city had been broken.
But he added that, although the Shias who had been repressed by Saddam Hussein's regime were welcoming the marines as agents of change, the US is not popular in Iraq.
Other key developments:
- A spokeswoman for the International Committee of the Red Cross says the disorder means it is too dangerous to work in Baghdad. One Canadian Red Cross worker is missing after his car was hit by gunfire on Tuesday.
- British soldiers take the first steps to restore order to the southern Iraqi city of Basra as looting begins to subside.
- Journalists with the Arabic television stations Al-Jazeera and Abu Dhabi TV are trapped overnight in the offices of Abu Dhabi TV by fighting between US and Iraqi forces
- Two US airmen are missing after their F-15 fighter jet went down over Iraq on Sunday, the US Defence Department says
- Two injured US special forces soldiers are rescued near Baghdad and taken to hospital in Kuwait, Central Command announces
The Americans have been advancing from three directions, extending the central area they have held since Tuesday morning.
Children helped parents carry goods away
A US military spokesman says that almost an entire armoured brigade - several thousand troops - is now in Baghdad, and it intends to stay there.
And the Americans say they are planning to double the size of the force in Baghdad over the next 24 hours.
The US military also says its warplanes can fly anywhere over Iraq with impunity.
"Coalition air forces have established air supremacy over the entire country - which means the enemy is incapable of effective interference with coalition air operations," said Pentagon spokesman Major-General Stanley McChrystal.
However, there is continuing uncertainty among coalition officials about whether Saddam Hussein was caught in a specially-targeted US bomb attack in Baghdad on Monday.
British security sources told the BBC they believed the Iraqi leader escaped the attack by several minutes.
US tanks were cheered as they pushed into central Baghdad
An unnamed US military intelligence source says the CIA believes he was killed, the Washington Times reports.
Outside Baghdad, coalition forces have made a number of advances.
US forces in central Iraq have captured the division headquarters of the Iraqi 10th armoured division without a fight after the division's 15,000 troops vanish, the BBC's Andrew North reports.
Kurdish sources say Kurdish fighters backed by US special forces in the north have taken control of a strategically important mountain near the city of Mosul.
US marines in eastern Iraq are close to linking up with UK troops coming from Basra in the south, opening an eastern supply corridor to Baghdad after taking positions without resistance on Tuesday, a BBC correspondent says.