Official projections showed the right-wing New Democracy and Pasok parties winning 163 seats, enough for a majority.
New Democracy leader Antonis Samaras, who said Greeks had voted to stay in the euro, said he wanted to form a coalition government as quickly as possible.
Alexis Tsipras, the leader of the anti-bailout Syriza party, which came a close second, conceded defeat and agreed Mr Samaras should be first to try to form a coalition.
Eurozone governments held "intense" discussions on the situation in Greece as voters went to the polls. European leaders warned that if a Greek government rejects the bailout, the country could be forced out of the single currency.
Speaking in central Athens, Mr Samaras told supporters: "I will make sure that the sacrifices of the Greek people will bring the country back to prosperity.
"We are determined to do what it takes, and do it fast. I will work with everyone in order to achieve our national goal to come out of this crisis with social unity, jobs and security -for every Greek."
The White House said last night it hoped the election result would lead quickly to a new government that makes "timely progress" on economic challenges.
It said it was in the interests of all for Greece to remain in the eurozone and the US would engage with the European country in the spirit of partnership.
With 60% of the votes counted, projections last night put New Democracy on 30.1% of the vote, Syriza on 26.5% and Pasok on 12.6%.
Mr Samaras said Greeks "have voted for a European future for Greece", adding: "There will be no doubt about the position of Greece in Europe. Our people have honoured us with their vote and I would like from my heart to thank everyone."
Pasok leader Evangelos Venizelos said: "I am relieved. I am relieved for Greece and Europe. As soon as possible we will form a government."
Mr Tsipras, however showed little sign of political accord, adding: "We propose to upset the austerity measures and the bailout."
Supporters of Greece's extreme right Golden Dawn party were also celebrating the election results. It is projected that the party will win enough votes to enter parliament on the back of anti-immigrant sentiment and concern about security.
Germany's Foreign Minister Guido Westerwelle said yesterday Greece must stick to its agreements with international creditors – but suggested Athens might be given more time to comply with them.
"There cannot be substantial changes to the agreements, but I can well imagine talking again about timelines," he said.
Meanwhile Tony Blair questioned the survival of the euro unless growth is linked to fundamental reform in Europe. As prime minister, Mr Blair wanted to take Britain into the euro.
He said the only long-term solution was for Germany to stand fully behind the single currency. But he said the Germans would be prepared to do this only if wide-ranging reforms were also delivered by European governments.
Mr Blair, who left office in 2007, said: "The real difficulty you have got in Europe at the moment is that people are being offered a choice between, on the one hand, austerity plus major structural reform, and, on the other hand, policies of growth and no reform.
"Actually, the only way you are going to get through this is to have a combination of policies that promote growth and at the same time governments undertaking the deep structural reform Europe needs.
"This is the really difficult thing from Germany's point of view ... what we need in order for the single currency to survive is Germany to come fully behind the single currency."