Earlier Sunday, several thousand people rallied in Tblisi, claiming the vote was rigged by the government of Saakashvili, who was once praised as an icon of democracy in the former Soviet Union but shocked his Western allies when he crushed anti-government demonstrations late last year and shut down an independent television station.
The influential election observer mission of the Organization for Security and Cooperation in Europe said the election revealed "significant challenges" but was generally in line with democratic commitments.
The OSCE pointed especially to "an inequitable campaign environment" due to state activities overlapping Saakashvili's campaign.
The vote count had trickled in slowly; hours earlier election officials said only about 30 percent of precincts had been counted.
Opposition leaders said the campaign was unfair and alleged widespread voting violations.
Addressing a crowd of some 5,000 people in a snowy square in Tbilisi, the opposition candidate Levan Gachechiladze claimed he came first in the vote and called for a second round. He cited a tally by his supporters who served on election commissions across the country.
"Saakashvili lost, and it cannot happen that Georgia will not defend it's freedom, that we won't win," Gachechiladze said.
The two-hour rally ended peacefully.
Nino Burdzhanadze, the parliament speaker who is serving as acting president during the campaign, conceded that there were some violations, but said her government welcomed the criticism from foreign observers and would analyze and seek to correct the mistakes in future elections.
"What is most important is that, in general, as a whole, the elections were free and fair and democratic," Burdzhanadze told The Associated Press. "You should take into account that Georgia is a new democracy ... and of course we still need to continue to develop democratic institutions, to develop democracy in the country."
Russia, which imposed an economic blockade on Georgia after repeated disputes with Saakashvili, was quick to criticize the president.
"The indignation of the opposition supporters over (Saakashvili) essentially declaring himself to be the victor, without waiting for official preliminary results, is completely understandable," the Russian Foreign Ministry said in a statement.
Gachechiladze said the opposition would contest the election results in courts, but will return to the streets if their efforts prove futile. He called for another rally Tuesday.
Saakashvili's supporters poured onto the streets late Saturday, tooting car horns and waving white-and-red national flags, celebrating victory based on exit poll results.
While still waiting for official results, Saakashvili said the exit poll showed him winning and called for reconciliation.
"I'm extending my hand to those who voted for me and to those who took part in the elections," he told supporters.
After voting in Tbilisi, Saakashvili said he was dedicated to having a free and fair election. "We are committed to having Georgia as a beacon of democracy in our part of the world," he said.
Gachechiladze, 43, represents an opposition coalition that wants to do away with the presidency. If a parliamentary system is established, as the coalition wants, he would step down.
The opposition has been undermined by a scandal that has discredited one of the leading candidates, billionaire Badri Patarkatsishvili. He has been accused of plotting to overthrow the government and has acknowledged offering large sums of money to police if they side with protesters. Georgian authorities say Patarkatsishvili - running a distant third in the exit poll - offered $100 million.
During his four years in office, Saakashvili has cracked down on organized crime and corruption, modernized the police force and the army, restored steady supplies of electricity and gas, and improved roads. The result has been annual economic growth of about 10 percent and a steady rise in foreign investment.
The economic success has not yet defeated poverty, and after the November protests, Saakashvili made social welfare one of his top priorities.
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Associated Press writers Maria Danilova and Lynn Berry contributed to this report.
