Antonis Samaras said Greeks had voted to stay in the euro, and called for a "national salvation government".
The leader of the anti-bailout Syriza party, Alexis Tsipras, which came a close second, agreed Mr Samaras should be first to try to form a coalition.
Germany said it viewed the result as a decision to "forge ahead" with reform.
With 80% of votes counted, interior ministry projections put New Democracy on 29.9% of the vote (130 seats), Syriza on 26.7% (71) and the socialist Pasok on 12.4% (33).
New Democracy leader Mr Samaras said: "The Greek people voted today to stay on the European course and remain in the eurozone.
"There will be no more adventures. Greece's place in Europe will not be put in doubt."
"The sacrifices of the Greek people will bring the country back to prosperity," he promised.
He also said Greece would "honour its obligations".
The BBC's Mark Lowen in Athens says that suggests Mr Samaras wants to press ahead with spending cuts demanded by the country's international creditors.
European leaders have warned that if a new Greek government rejected the bailout, the country could be forced to abandon the single currency.
While the radical-left Syriza and other smaller parties have opposed the bailout, New Democracy and Pasok said they would keep it in a renegotiated form.
Germany's Finance Minister, Wolfgang Schauble, said he viewed the election result as a decision by the Greek people "to forge ahead with the implementation of far-reaching economic and fiscal reforms in the country".
Eurozone finance ministers said in a statement that such reform was "Greece's best guarantee to overcome the current economic and social challenges and for a more prosperous future [for] Greece in the euro area".
They said they expected representatives of the European Commission, the European Central Bank and the International Monetary Fund - the so-called Troika - to return to Athens as soon as there was a Greek government in place.
Germany Foreign Minister Guido Westerwelle suggested Athens might be given more time to comply with its obligations.
"There cannot be substantial changes to the agreements, but I can well imagine talking again about timelines," he said.
Mr Tsipras, the Syriza leader, congratulated Mr Samaras on his apparent victory, and said he had the right to try to form a government.
But he appeared to rule out joining such a coalition, saying Syriza would "not sacrifice our position" of opposition to the austerity programme.
If the projections from the interior ministry are proved correct, New Democracy should be able to build a majority coalition with the socialist Pasok, benefiting from a rule which gives the leading party 50 extra seats in the 300-seat chamber.
However, coalition talks may not be easy. After a first, inconclusive election six weeks ago, each of the main parties tried but failed to form a coalition government.
The leader of Pasok, Evangelos Venizelos, proposed a broad four-party coalition including New Democracy, Pasok, the Democratic Left and Syriza.
"No decision can be taken without this national unity," he said.
Analysts suggested Mr Venizelos had doubts over the viability of a coalition with a narrow majority.
When his party was in power it suffered numerous defections and rebellions as it tried to impose unpopular austerity measures.
Our correspondent, Mark Lowen, says that with such a strong showing by Syriza, Greece could be in for an autumn of discontent by opponents of the bailout deal.
Four other anti-bailout parties look set to take between 60 and 70 seats. They include the far-right Golden Dawn, which looked set to secure about 7% of the vote.
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 deepen the turmoil in the global economy.
Tough austerity measures were attached to the two international bailouts awarded to Greece, an initial package worth 110bn euros (£89bn; $138bn) in 2010, then a follow-up last year worth 130bn euros.
Polls suggest most Greeks want to be rid of the bailout and its onerous conditions, but want to stay in the euro.