"Those will be tough negotiations".
For their part, the nationalists sought to join forces and form a government.
Tomislav Nikolic, the acting leader from the Radical party, met Vojislav Kostunica, the outgoing nationalist prime minister, on Monday to see if they could form a government together.
"There is a clear chance that a government will be formed that will not include Tadic's party," Nikolic said.
Any alliance that can produce a 126-seat majority in the 250-seat parliament can govern.
Tadic's coalition appeared assured of 103 seats while Nikolic's Radicals were poised to get 76.
If the Radicals joined forces with Kostunica's bloc and the Socialists, their combined strength would be 127 seats.
Andreja Mladenovic, Kostunica's spokesman, said he would hold talks in the coming days with other parties, including late president Slobodan Milosevic's Socialist party, whose support looked to be decisive for either camp.
Kosovo on agenda
The Socialists said they would start talks with Kostunica's party on Tuesday.
"At the moment, everyone is expecting Kostunica and the Radicals and us to form a coalition, but we still haven't talked," Branko Ruzic, a vice-president of the Socialists party, told Al Jazeera.
"We will also talk to Mr Tadic because he won the election.
"What will be the outcome, I'm not sure, but ... we need to have a government that keeps the national interests of Serbia.
"We need to have guarantees that no one will give up Kosovo for a European future that we cannot see," he said.
Nikolic also accused Tadic of inciting violence by proclaiming victory.
But Tadic made clear he saw the outcome as a mandate to take the country into the EU.
He warned his opponents "not to tamper with the will of the people'" and pledged to prevent the formation of a nationalist government.
Tadic also was expected to court the Socialists and their 21 seats.
The European Union called the success of Tadic's coalition a "clear victory" by pro-European forces.
The US embassy in Belgrade said in a statement that "the Serbian electorate has clearly demonstrated that its heart is in Europe".
"Serbia's citizens have spoken out in favour of a prosperous future inside the Western community," it said.
Carl Bildt, the Swedish foreign minister, said pro-European parties have won an "important moral victory".
But he cautioned that "there is no altogether unambiguous majority. The manoeuvring can take a great deal of time".
Near-complete official results released on Monday corresponded to projections by the Centre for Free Elections and Democracy and tabulations by the main parties.
The institute, whose representatives observed vote tallying at polling stations across Serbia, said Tadic's bloc had 39 per cent.
It said the Radicals ran a distant second with 28.6 per cent, and Kostunica's bloc had about 11.6 per cent.
The Socialists had about 8.2 per cent - their best result since Milosevic's ouster in 2000.