Before the vote, opinion polls had suggested Tomislav Nikolic's Radical Party was running neck and neck with Mr Tadic's Democratic Party.
Kosovo's declaration of independence was considered a major campaign issue.
Disagreements over how to react when most EU countries recognised Kosovo had brought down the previous coalition between the Democratic Party and Prime Minister Vojislav Kostunica's nationalist Democratic Party of Serbia.
Some seven million Serbs were eligible to go to the polls and the turnout was 60.7%, according to the non-governmental Centre for Free Elections and Democracy.
Sigh of relief
Mr Tadic said the Serbian people had in effect given their approval to the country's membership of the EU, with whom a long-delayed agreement was signed just before the election.
PROJECTED ELECTION RESULTS
Tadic bloc: 38.7%, 103 seats
Radicals: 29.1%, 77 seats
Kostunica bloc: 11.3%, 30 seats
Socialist bloc: 7.9%, 20 seats
Liberal Democrats: 5.2%, 13 seats
Ethnic minorities: 2.3%, seven seats
Total seats: 250. Parties forming a coalition need to have at least 126 between them
Source of figures: independent monitoring group the Centre for Free Elections and Democracy
"The people of Serbia have undoubtedly confirmed that Serbia is clearly on a European path," he told supporters in Belgrade.
A statement from the EU's Slovenian presidency welcomed a "clear victory" for pro-European forces, and said it hoped they would quickly form a government.
A projection of the result based on a sample count throughout the country suggested the Democratic Party and its allies had won about 39% and the Radicals, 29%.
Mr Tadic's alliance is projected to have around 103 seats in the 250-seat parliament.
He could move to form a coalition with the Liberal Democrats, the only Serbian party that accepts the independence of Kosovo. However it was not immediately clear if they had passed the 5% threshold for entering parliament.
Serbs celebrating on the streets of Belgrade
Mr Tadic may have to turn to the most unlikely of bedfellows, the Socialist Party of the late dictator, Slobodan Milosevic, the BBC's Oana Lungescu reports.
It seems unlikely Mr Kostunica's party would enter a new coalition with Mr Tadic's supporters, given the bitterness of their earlier divisions.
However, theoretically, on the basis of the projections, the Radicals could form a coalition with Mr Kostunica and the Socialists, netting a wafer-thin majority in parliament.
President Tadic's victory, if confirmed, will trigger a collective sigh of relief across the EU, our correspondent adds.
The Radicals have not given up hope of forming a government
The Radicals' leader accused President Tadic of spreading fear on Sunday night by insisting his party was the only one capable of forming a new coalition.
"By harshly violating the constitution this evening, Boris Tadic indicated that civil unrest was possible," Mr Nikolic said.
Meanwhile, Tadic supporters have been celebrating in Belgrade with fireworks.
Cars decked with party and EU symbols blew their horns as they circled the city centre.
Serbs in Kosovo also voted, both in the general election and in council elections, in defiance of the ethnic Albanian government and international authorities.
Some 300 polling stations opened in areas with a sizeable Serb population and the vote passed off peacefully.
However, even before the polls closed, the United Nations mission issued a statement, condemning the council elections as illegal and saying the results would not be recognised.
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