Independent monitors said the alliance led by the Democratic Party had won 39% of the vote, ahead of the nationalist Radical Party, which got 29%.
The rolling count by the state election commission put the Democrats at 36.7% the Radicals at 28.5%.
The election was fought on whether Serbs should swallow their anger over EU support for the independence of Kosovo, the Serb province that seceded in February, or turn their backs on the bid for EU membership.
The Democrats celebrated in the streets, and their leader, President Boris Tadic, said: 'Serbs have undoubtedly confirmed a clear European path.
'This is a great victory, but it's not over yet,' Mr Tadic said. 'I want us to be aware that we must form a new government as soon as possible.'
The Radicals' leader, Tomislav Nikolic, said the Democrat claim of victory jumped the gun.
'There are very clear possibilities of a coalition which does not include the Democratic Party', he said.
Mr Nikolic said he would talk to the two parties that share the Radicals' ideology, the Democratic Party of Serbia led by outgoing nationalist premier Vojislav Kostunica and the Socialists of the late Slobodan Milosevic.
Either these three parties would form a coalition, he predicted, or 'Serbia will not have a government at all and we'll have to go to new elections'.
Official results are due by Thursday night. Parliament must convene by mid-June and a government be formed by mid-September or the country must hold a new election.
