Alexis Tsipras, leader of the communist-rooted Syriza party, immediately promised to end the "five years of humiliation and pain" that Greece has endured since an international bailout saved it from bankruptcy in 2010.
With 80 per cent of polling stations counted, Syriza had 36 per cent versus 28 per cent for Prime Minister Antonis Samaras' conservatives.
It remained to be seen whether Syriza had enough seats to govern outright or would have to seek support from other parties. That might not become clear until Monday morning or even later, when all the votes are counted.
If Tsipras, 40, can put together a government, he will be Greece's youngest prime minister in 150 years.
The prospect of an anti-bailout government coming to power in Greece has sent jitters through the financial world, reviving fears of a Greek bankruptcy that could reverberate across the eurozone.
"The sovereign Greek people today have given a clear, strong, indisputable mandate. Greece has turned a page. Greece is leaving behind the destructive austerity, fear and authoritarianism. It is leaving behind five years of humiliation and pain," Tsipras told a crowd of rapturous flag-waving supporters.
He won on promises to demand debt forgiveness and renegotiate the terms of Greece's 240 billion-euro ($270 billion) bailout, which has kept the debt-ridden country afloat since mid-2010.
To qualify for the cash, Greece has had to impose deep and bitterly resented cuts in public spending, wages and pensions, along with public sector layoffs and repeated tax increases.
Its progress in reforms is reviewed by inspectors from the International Monetary Fund, European Commission and European Central Bank, collectively known as the troika, before each installment of bailout funds can be released.
Tsipras pronounced the troika and its regular debt inspections "a thing of the past."
"The verdict of the Greek people ends, beyond any doubt, the vicious circle of austerity in our country," he said.
Greece's creditors insist the country must abide by previous commitments to continue receiving support.
The election results will be the main topic at Monday's meeting of eurozone finance ministers. Belgium's minister, Johan Van Overtveldt, said there is room for some flexibility, but not much.
"We can talk modalities, we can talk debt restructuring, but the cornerstone that Greece must respect the rules of monetary union — that must stay as it is," Van Overtveldt told VRT network.
Samaras conceded defeat, saying he had received a country "on the brink of disaster" when he took over in 2012 and was close to ushering it out of the crisis.
"I was asked to hold live coals in my hands and I did," he said. "Most gave us no prospect of lasting out and we did. We got the country out of deficits and recession ... and set the foundations for growth and a final exit from the crisis."
Syriza's anti-bailout rhetoric appealed to many in a country that, in the past five years, has seen a quarter of its economy wiped out, unemployment above 25 per cent, and average income losses of at least 30 per cent.
But Syriza's victory has renewed doubts over Greece's ability to emerge from the crisis, and generated fears that the country's finances could once again send shockwaves through global markets and undermine the euro, the currency shared by 19 European countries.
The centrist Potami party was battling for third place with the Nazi-inspired Golden Dawn, whose leader and several lawmakers campaigned from prison, where they are awaiting trial on charges of participating in a criminal organization.
If Syriza falls shy of the 151 seats necessary to form a government on its own in the 300-seat parliament, it will have to seek support from other parties — either in a minority government or as part of a coalition.
Raf Casert in Brussels and Derek Gatopoulos in Athens contributed.