The Georgian Dream coalition, headed by billionaire Bidzina Ivanishvili, has claimed victory but so has the party of current President Mikheil Saakashvili.
However, the electoral system means that even if final results bear out the exit polls, the opposition may not have achieved a parliamentary majority.
Official results are due on Tuesday.
Supporters of Georgian Dream have already begun gathering to celebrate in the capital Tbilisi.
"We have won! The Georgian people have won!" Mr Ivanishvili said in a speech broadcast on a Georgian TV station, the AFP news agency reports.
Early exit polls showed the opposition has a clear lead in votes for party lists, which determine 77 out 150 parliamentary seats.
In comments shown on Georgian TV, Mr Saakashvili conceded the opposition was ahead in that race but said his United National Movement was leading in the race for the 73 seats elected on a first-past-the-post basis.
A spokeswoman for the United National Movement predicted that it would keep its majority.
Earlier Mr Ivanishvili, who Georgia's richest man, had staged a symbolic protest by refusing to vote, saying the authorities had "already resorted to very many violations".
The poll is being seen as the biggest test of President Saakashvili's popularity since he came to power in 2003.
Georgia's political system of different-sized constituencies means it is possible to win more seats in parliament, but with fewer votes in total, the BBC's Damian McGuinness in Tbilisi reports.
If the ruling party gets back into power that way, the opposition could well feel cheated of victory - and spark mass protests, our correspondent adds.
The government's reputation has taken a battering in recent weeks because of a prisoner-abuse scandal.
Videos broadcast on national television showed prison inmates being beaten and sexually abused by guards.
The scandal sparked street protests and allowed Mr Ivanishvili to portray the government as high-handed and uncaring.
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