New Democracy, which broadly supports a European bailout deal, was one to two percentage points ahead of Syriza, which opposes the measure.
The outcome could decide Greece’s future inside the euro.
If the exit poll is correct, New Democracy should be able to build a majority coalition.
The government will be relatively weak, and will seek to change the terms of the bailout.
The election was the second in six weeks, called after a 6 May vote proved inconclusive.
On that occasion, each of the main parties tried but failed to form a coalition government.
New Democracy is thought to have polled between 28 and 30% of votes, with 27-28% for Syriza, one exit poll carried out jointly by five polling companies for the main TV channels showed.
An earlier version of the poll only 80% complete had the two parties virtually neck and neck, prompting fears of a hung parliament.
The latest projection would give New Democracy 127 seats, benefiting from a rule which gives the leading party 50 extra seats in the 300-seat chamber.
It gave the centre-left Pasok, its potential coalition partner, 32 seats, enough for a majority in the 300-seat parliament, with Syriza gaining 72 seats.
New Democracy could also invite a small left-wing party, Democratic Left, to join the coalition to reflect some of the anti-bailout feeling in the country.
With such a strong showing by Syriza, Greece could be in for an autumn of discontent by opponents of the bailout deal.
Another poll for a separate TV station gave Syriza a marginal lead.
Coalition talks will be expected to start on Monday.
Several smaller anti-bailout parties are expected to get between 13 and 21 seats.
Sunday’s vote is being watched around the world, amid fears that a Greek exit from the euro could spread contagion to other eurozone members and send turmoil throughout the global economy.
Tough austerity measures were attached to the two international bailouts awarded to Greece, an initial package worth 110 billion Euros ($138 billion) in 2010, then a follow-up last year worth 130 billion Euros.
Many Greeks are unhappy with the conditions attached to deals which have been keeping the country from bankruptcy.
Polls have shown most Greeks favor staying in the euro and all the main parties except the communist KKE say they will keep Greece in the single currency, but Syriza believes it can renegotiate the bailout deal.
New Democracy leader Antonis Samaras has warned that rejection of the bailout would lead to a return to the drachma, but correspondents say a very large number of Greeks appear to have defied this pressure.
“Greeks voted with emotion and not with reason, this is why you see these numbers,” New Democracy supporter Evangelos Datsos told Reuters news agency after the initial exit polll results came through.
But Syriza supporters were confident of victory.
There is a subdued atmosphere in the Greek capital on Sunday night, with many people just at home, watching nervously on television.
Greeks are proud and therefore private when it comes to explaining their fears to foreigners, our correspondent says, but behind closed doors they are worried about what this means for their country and their futures.
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