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Sporadic skirmishes between Mr Kostunica's supporters and security forces across Belgrade have been light and short-lived.
Demonstrators also ransacked the headquarters of the propaganda mouthpiece of the Milosevic regime, Radio Television Serbia, and set the building on fire forcing three state channels off the air.
Independent radio station B2-92 has announced that senior army officials have made contact with the opposition and withdrawn their forces from the centre of the capital.
The relatively peaceful revolution began with a rally this morning to contest the legitimacy of President Milosevic's regime and his efforts to maintain his authority.
A state radio broadcast just before 1100 local time (0900 GMT) announced the Constitutional Court had annulled last month's presidential election and ordered a re-vote, effectively giving Mr Milosevic another year in power.
But opposition leaders in turn gave Milosevic an ultimatum to surrender power by 1500 (1300 GMT). They then filed criminal charges against him for vote rigging.
In the early afternoon police fired tear gas into the huge crowds of Kostunica supporters surrounding the state parliament.
Protestors were temporarily halted at the steps of the building before police withdrew. Some officers even joined the demonstration themselves.
Using a bulldozer to gain entry, groups then made their way inside, lighting fires and smashing furniture and computer equipment.
Britain and the US have given their support to the popular uprising and have urged Mr Milosevic to stand down.
President Bill Clinton said the US stood by those who were "fighting for their freedom". Meanwhile the Yugoslav state news agency Tanjug has asserted its independence from state control.
Mr Kostunica is expected to address a vast crowd assembled in front of the Yugoslav parliament shortly.
If Russia gives its blessing and the Yugoslav army officially recognises Mr Kostunica's legitimacy he could be sworn in as president by the end of the week.
Slobodan Milosevic announced his resignation in a televised address to the nation the following day.
Mr Kostunica pledged to re-establish normal relations with the rest of the world. Sanctions against Yugoslavia for Serbian aggression in Kosovo were almost immediately lifted.
Yugoslavia is currently receiving structural aid in return for its co-operation with the UN Security Council, such as the extradition by Yugoslav authorities of Slobodan Milosevic to the Hague in 2001.
Mr Milosevic is the first former head of state to be tried at the International Criminal Court in the Hague.
He was put on trial for war crimes in Kosovo, Croatia and Bosnia but died before the trial ended in March 2006.
Some 250,000 people died in the 1992-1995 Bosnian war and about 20,000 people lost their lives in the 1991 Croatian war.
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