The  
          eurozone group
 said reforms were Greece's "best guarantee" to overcome tough economic and social challenges.
European markets were mixed in morning trading, after Asia had recorded earlier gains.
The Syriza party, which rejected the bailout terms and came a close second, said it would lead the opposition.
With 99.9% of ballots counted, interior ministry results put New Democracy on 29.7% of the vote (129 seats), Syriza on 26.9% (71) and the socialist Pasok on 12.3% (33).
There are 300 seats in parliament and Greece has a rule that gives the leading party 50 extra seats.
Greek voters had gone to the polls on Sunday following inconclusive elections in May.
New Democracy leader Antonis Samaras said Greeks had chosen to stay in the euro and called for a "national salvation government".
By Gavin HewittEurope editor
If Antonis Samaras is confirmed in power he will have a stronger opposition breathing down his neck. He will say to Europe that he has anchored Greece in the eurozone and wants something in return. That will be difficult.
Already German Foreign Minister Guido Westerwelle has come out and said that the substance of the Greek reform bailout programme is non-negotiable. Where there might be some flexibility is over the timing of the implementation of reforms. There may also be some movement over interest rates on the loans and the EU might offer Greece some funds to boost growth.
But the message is clear; the austerity programme with its budget and spending cuts will stay.  The German finance minister underlined that message when he said: "Greece's path will be neither short nor easy".
He is now meeting President Carolos Papoulias, who is expected to give him the mandate to try to form a government.
Syriza's leader Alexis Tsipras said his party would not take part in the government, and would instead become a powerful anti-austerity voice in the opposition.
The BBC's Mark Lowen, in Athens, says Mr Samaras is expected to try for a broad coalition, hoping to create a stable government with a stronger popular mandate.
He says Mr Samaras will push for a lightening of the bailout terms from Brussels, arguing that Greeks have accepted more pain by electing a pro-bailout party and that Europe should now cut Greece some slack.
However, correspondents also point out that only 40% of voters backed parties that broadly support the bail-out deal with the EU and the IMF.
Tough austerity measures were attached to the two international bailouts awarded to Greece, an initial package worth 110bn euros (Â£89bn; $138bn) in 2010, then a follow-up last year worth 130bn euros.
Sunday's vote was 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.
European stock markets were mixed in morning trading. Asian shares had earlier advanced. Japan's Nikkei 225 index and South Korea's Kospi both closed up 1.8%.
BBC business editor Robert Peston says the election results have been seen by investors as avoiding the worst short-term outcome, which would have been political paralysis followed by a messy exit from the euro.
In a statement on behalf of the 17 eurozone finance ministers, Luxembourg Prime Minister Jean-Paul Juncker said that "continued fiscal and structural reforms are Greece's best guarantee to overcome the current economic and social challenges and for a more prosperous future of Greece in the euro area".
Election as it happenedProfile: Antonis Samaras In pictures: Greek elections
In  
          a joint statement
 European Commission President Jose Manuel Barroso and European Council President Herman Van Rompuy also hailed the result, and expressed hope that a government would be formed quickly.
German Chancellor Angela Merkel telephoned Mr Samaras to congratulate him on his victory.
A German government statement said she had "stated that she would work on the basis that Greece will meet its European commitments".
The US stressed that it was in everyone's interests "for Greece to remain in the euro area".
Mr Samaras said that the Greek people had voted to stay in the eurozone.
"There is no time to waste," he said. "A national salvation government must bring economic growth and reassure Greeks the worst is over."
He added: "There will be no more adventures. Greece's place in Europe will not be put in doubt," promising that Athens would "honour its obligations".
Our correspondent, Mark Lowen, says this suggests that Mr Samaras wants to press ahead with spending cuts demanded by the country's international creditors.
European leaders have warned that if the new Greek government rejects the bailout, the country could be forced to abandon the single currency.
New Democracy should be able to build a majority coalition with the socialist Pasok.
However, coalition talks may not be easy.
In addition to Syriza's showing, four other parties which oppose or want a radical overhaul of the bailout look set to take between 60 and 70 seats. They include the far-right Golden Dawn, which has about 7% of the vote.
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.
World leaders have welcomed the narrow election victory of Greece's broadly pro-bailout New Democracy party and urged Athens to form a cabinet quickly.