With most districts counted, Prime Minister Alexis Tsipras admitted defeat to his rival, Kyriakos Mitsotakis.
New Democracy has 39.8% of the vote so far, with Mr Tsipras's leftist Syriza party in second place with 31.6%.
Current projections give New Democracy an outright majority, as the winner receives 50 extra seats in parliament.
So far about 88% of districts have returned their results.
The prime minister-elect told supporters he had been given a strong mandate for change.
"The country proudly raises its head again," he told the crowd in the capital Athens, saying he would be a president for all because Greeks were "too few to stay divided".
Speaking as the results became clear, Mr Tsipras confirmed he had called Mr Mitsotakis to offer him his congratulations.
"Today, with our head held high we accept the people's verdict. To bring Greece to where it is today we had to take difficult decisions [with] a heavy political cost," Mr Tsipras told journalists.
Turnout in the election was about 57% – one of the lowest figures in decades.
Young Greeks lean to conservatives for future
Mitsotakis and the return of a Greek dynasty
Voting is technically compulsory in the country, though the rule is not enforced.
There have been a half dozen elections in recent years – and on Sunday, many residents flocked to the beaches or sheltered at home as temperatures exceeded 35C in places.
What do results say so far?
Early results also predicted the far-right Golden Dawn party would be short of the 3% minimum needed to enter parliament by the narrowest of margins - at 2.95% for now.
The nationalist pro-Russian Greek Solution and MeRA25, the left-wing party of former Finance Minister Yanis Varoufakis, were both projected to just exceed the electoral threshold.
The pendulum swings back
On the streets of Athens, there was a muted reaction as the results were announced. New Democracy's win was expected, and the crowds were at beaches rather than rallies.
After its sixth election in a decade, this tired country seemed to give a collective shrug.
In 2015, New Democracy was seen as the corrupt old guard, blamed for leading Greece into the crisis as it swapped power with the Socialists over decades. The leftist, populist Alexis Tsipras represented change: vowing to rip up the bailout and austerity.
But he was undone by the humiliating U-turn forced by the EU, as well as perceived mismanagement and scandals.
And some flirting with populism, from the UK to the US, will see in Greece that the political pendulum can swing back to the mainstream.
The centre-left Movement for Change is on 7.9%, followed by the Communist Party on 5.4%.
Greek media also report a ballot box was stolen by an unidentified group which had forced its way into one polling station.
What happened to Syriza's support?
Prime Minister Tsipras, who swept to power in 2015, called the elections after suffering defeat in the local and European votes in May.
He had toppled New Democracy from power by campaigning against the austerity policies that had affected Greeks for years.
Yet within months, he was forced to accept tough conditions in return for Greece's third international bailout.
Mr Tsipras faced a revolt from within his party as a result and responded by calling fresh elections, which he won.
However, his party's popularity was dented as unemployment increased, and Greece's economy shrank.
Media captionWhy young people have suffered the most from Greece’s economic collapse
Mr Mitsotakis attacked his rival's record, saying that economic improvement was too slow.
He also benefited from widespread anger at a landmark deal with North Macedonia over the name of the country - which it shares with a Greek region.
What does New Democracy stand for?
New Democracy has promised to lower taxes and privatise services in the country, which is still feeling the effects of the 2008 financial crisis.
Greece has been in receipt of a series of bailout programmes over the past decade, which it officially "exited" last August as economic growth returned.
But youth unemployment remains high, and New Democracy has counted many 18-24 year olds among its supporters.
The potential new prime minister comes from one of Greece's long-standing political dynasties.
Kyriakos Mitsotakis is the son of former Prime Minister Konstantinos Mitsotakis. His sister, Dora Bakoyannis, was mayor of Athens when the city hosted the Olympics in 2004, before becoming Greek foreign minister.