Rebel commanders say they have taken control of about 80% of the capital, including the headquarters of state TV.
But fighting is still raging in parts of the city, and the rebels have not managed to find the Libyan leader.
World leaders have urged Col Gaddafi to step down; Egypt has recognised the rebels as the legitimate government.
The rebels were met by jubilant crowds in central Green Square, which was previously the scene of nightly pro-Gaddafi demonstrations.
They have set up checkpoints in parts of the city, and claim that reinforcements are due to arrive by boat.
But elsewhere in the city they have met stiff resistance.
A rebel spokesman said his forces came under fire from tanks emerging from the Gaddafi compound at Bab al-Azizia in western Tripoli early on Monday, and witnesses say there has been sustained gunfire in the area throughout the day.
Gaddafi loyalists remain in control of the streets around the Rixos Hotel, where many Western journalists are based.
And rebels fighting in the west of the city were pushed back late on Monday.
"We are bracing ourselves for another night of intense street fighting," a Tripoli resident told the BBC.
"I think Gaddafi security forces will resort to guerrilla warfare because they know they do not have the support of the people."
But another resident said rebel fighters were "breaking into people's houses, stealing everything", adding that the rebel assault would be "a disaster for Libya and Nato".Continue reading the main story Rebel leader Mustafa Abdul Jalil, head of the National Transitional Council (NTC), told a news conference on Monday afternoon that he had no idea where Col Gaddafi was.Continue reading the main story David Loyn BBC International Development Correspondent
The unity of the rebel forces will come under severe test in the coming days because of the imbalance of power in the country. All of the key military advances have come from forces in the west. But political power is in the hands of leaders from the east in the National Transition Council.
Libya's rebels have in their advance on Tripoli shown far better co-ordination than in the past, and their desire to put members of Gaddafi's family on trial shows restraint and not revenge.
Medical and food aid are the first priority. As soon as possible, western embassies in Tripoli will reopen, and stabilisation advisers, building on lessons learnt in Afghanistan, will assist in ensuring that government services continue.
A draft transition document has an ambitious timetable for a constitution to be drawn up within three months and elections held six months after that. The document also guarantees freedom and basic human rights under the law."We have no knowledge of Gaddafi being [in his compound], or whether he is still in or outside Libya," he said.
Mr Abdul Jalil reiterated earlier claims that the rebels have captured Col Gaddafi's son, Saif al-Islam, who was widely expected to succeed his father as leader until the unrest broke out earlier this year.Flags torn down
The BBC's Tripoli correspondent, Rana Jawad, who has been unable to report openly since March, says people in her neighbourhood in eastern Tripoli were woken by the imam at the local mosque singing the national anthem of the pre-Gaddafi monarchy.
She says there is a sense that the end is near, and that the rebels have achieved what they wanted.
In Green Square rebel supporters tore down the green flags of the Gaddafi government and trampled on portraits of the colonel.
US President Barack Obama in a statement: "The momentum against the Gaddafi regime has reached a tipping point. Tripoli is slipping from the grasp of a tyrant."
UK Prime Minister David Cameron said the Libyan leader had "committed appalling crimes against the people of Libya and he must go now to avoid any further suffering for his own people".
Russia and China also issued statements saying they were willing to accept what the Libyan people decided, and hinted that Col Gaddafi should step down.
The rebels now claim to have detained three of Col Gaddafi's sons - Saif al-Islam, Muhammad and Saadi.Continue reading the main story 17 Feb: Anti-government protests begin across Libya24 Feb - 6 Mar: Rebels seize control of several towns and cities but are driven back17 Mar: UN authorises no-fly zone and "all necessary measures" to protect civilians19 Mar: First strikes by US, French and British warplanes halt pro-Gaddafi forces30 Mar: Foreign Minister Moussa Koussa defects16-25 Apr: Gaddafi forces bombard rebel-held MisrataMay-Jul: Stalemate as rebels and government troops engage in skirmishesMid-Aug: Rebels seize key towns of Zawiya and Gharyan outside Tripoli21 Aug: Rebels enter TripoliThe International Criminal Court in The Hague is negotiating the transfer of Saif al-Islam on charges of war crimes.
The court is also seeking the arrest of Col Gaddafi and the head of the intelligence service, Abdullah al-Sanussi.
Earlier, the NTC announced earlier that it would move its centre of operations to Tripoli from Benghazi, which has been in rebel hands since the early days of the uprising.'Armed gangs'
A diplomatic source told the AFP news agency that Col Gaddafi could still be in Bab al-Azizia. He has not been seen in public since May, although he has broadcast audio messages from undisclosed locations.
In an audio message broadcast late on Sunday, the Libyan leader urged residents to "save Tripoli" from the rebels.
"How come you allow Tripoli, the capital, to be under occupation once again?" he asked. "The traitors are paving the way for the occupation forces to be deployed in Tripoli."
Libyan Information Minister Moussa Ibrahim said the Gaddafi government still had 65,000 loyal soldiers under its command.
However, other reports claimed some forces have surrendered to the rebels, including the special battalion charged with securing Tripoli.
Mr Ibrahim said fighting in the city on Sunday had left 1,300 people dead and 5,000 wounded. There is no confirmation of the figures.
He accused Nato of backing "armed gangs" with air power, adding that the Gaddafi government was prepared to negotiate directly with the NTC.
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