While there are 23 candidates on the ballot, attention is focused on three: Giorgi Margvelashvili, representing the Georgian Dream coalition and handpicked by Prime Minister Bidzina Ivanishvili; Davit Bakradze, a member of Saakashvili's United National Movement; and Nino Burjanadze, the leader of the Democratic Movement-United Georgia party, Radio Free Europe reported.
Margvelashvili led in recent polls, but the Georgian Dream candidate threatened to drop out of the race if he does not receive the necessary 50 percent of the vote to avoid a runoff.
Constitutional reforms that concentrate most power with the prime minister will take effect after Sunday's election. Ivanishvili has said he will retire by the end of 2013, but has indicated who would be his successor, RFE said.
"What's important is that this vote proceeds today in a way that's peaceful, fair open and transparent, and so far, things seem to be moving in that direction," said Richard Norland, U.S. ambassador to Georgia.
Elections monitor Transparency International said the election has been "generally quiet," but the number of procedural violations observed increased over last year. The organization said a number of precincts don't have enough ballots.
Polling stations were set up abroad, including in Afghanistan, where about 1,500 Georgian troops are part of the NATO-led International Security Assistance Force, RIA Novosti said.
Georgians living in Russia can't vote, however, because Russia severed diplomatic ties with Georgia since the 2008 war over Georgia's breakaway republics of Abkhazia and South Ossetia.
After the war, Moscow recognized South Ossetia and Abhkazia as independent states. Few other nations followed suit and Tbilisi considers them Georgian territory.