Antonis Samaras, of the New Democracy party, on Monday met President Karolos Papoulias to be given a formal mandate.
Mr Samaras said he would seek changes in the terms of a bailout agreement reached with the EU and IMF.
The second-place Syriza party has rejected the terms of the bailout and said it would form the opposition.
Mr Samaras said as he met the president: “A national understanding by everybody is imperative.”
Under the constitution, Mr Papoulias has given Mr Samaras three days to form a government.
Mr Samaras said he believed he could form a working coalition.
Mr Papoulias said there was “a categorical imperative to form the government” immediately.
“The country cannot remain ungoverned for even an hour,” he said.
Mr Samaras met the leader of Syriza, Alexis Tsipras, after talking to the president.
Mr Tsipras said Syriza would remain in opposition and challenge the government, as his party had “different priorities”.
“The role of a strong and responsible opposition… is to intervene in a powerful way and this is what I assured Mr Samaras that we would do.”
However, he said Syriza would not be an obstacle to the formation of a government by New Democracy, which he said was in accordance with the mandate of the people.
Despite Mr Tsipras’s opposition, Mr Samaras said: “We need a national salvation government with as many parties as possible.”
Mr Samaras will meet the leader of the socialist Pasok party, Evangelos Venizelos, in the afternoon.
Those two parties should be able to form a majority coalition, but the BBC’s Mark Lowen in Athens says Mr Samaras wants the broader grouping to try to create a stable government with a stronger popular mandate.
With almost all ballots counted, New Democracy has 29.7% of the vote (129 seats), Syriza 26.9% (71) and Pasok 12.3% (33).
There are 300 seats in parliament and Greece has a rule that gives the leading party 50 extra seats.
However, correspondents point out that only 40% of voters backed parties that broadly support the bailout deal with the EU and the IMF.
Nevertheless, many world leaders hailed the election result.
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”.
German Chancellor Angela Merkel telephoned Mr Samaras to congratulate him on his victory, while the US said it was in everyone’s interests “for Greece to remain in the euro area”.
Our correspondent 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.
German Foreign Minister Guido Westerwelle said the substance of the bailout was non-negotiable, although the timeframe could be discussed.
However, another spokesman for the German government, Georg Streiter, said Greece should “stick to its agreements and fully implement the agreed reforms. Now is not the time for any kind of discounts to Greece”.
Although the market response to the election victory was mixed, European banking stocks fell sharply. Germany’s Commerzbank was down 3% and France’s BNP 2%, with analysts saying much uncertainty remained.