The vote is the second this year in Kosovo after Kurti's Vetevendosje Party fell short of a majority in February. Months of failed coalition talks prompted President Vjosa Osmani to dissolve parliament in November and call an early election.
Kurti's party was leading with 49.3 per cent of the vote, with 99 per cent of votes counted.
"Once the results will be certified we should swiftly constitute the parliament and then form immediately a new government," Kurti told a press conference at his party's headquarters.
"We don't have time to lose and we should move forward as fast as we can together."
He called on members of the opposition to support him in voting for international loan deals, which require a two-thirds majority to pass.
Kurti's supporters chanted his name as they celebrated in the streets of Pristina, waving his party's flags as fireworks lit the sky in temperatures of -3C.
Analysts say it is difficult to predict whether Kurti will be able to form the government on his own without a coalition to secure the 61 seats in the 120-seat assembly. Conditional votes and votes from the Kosovo diaspora in western European countries have not yet been counted.
"The results are not final and I don't see how Kurti will form the government alone but it will be very easy for him to govern with a small coalition," said Ismet Kryeziu of the Kosovo Democratic Institute think tank.
Kosovo's two main opposition parties, the Democratic Party and the Democratic League, were at 21 per cent and at 13.6 per cent, respectively.
Another failure to form a government and reopen parliament would prolong the crisis at a critical time. Politicians must elect a new president in April and ratify one billion euros ($A1.8 billion) in loan agreements from the European Union and World Bank that expire in the coming months.
The Balkan country's opposition parties have refused to govern with Kurti, criticising his handling of ties with Western allies and his approach to Kosovo's ethnically divided north, where a Serb minority lives. Kurti blames the opposition for the impasse.
In a bid to woo voters, Kurti has pledged an additional month of salary per year for public sector workers, one billion euros per year in capital investment and a new prosecution unit to fight organised crime. Opposition parties have also focused on improving living standards - a leading concern for voters.
Turnout was 45 per cent, according to the results.
Kosovo declared independence from Serbia in 2008 with US backing, including a 1999 NATO bombing campaign against Serbian forces trying to crush an uprising by the 90 per cent ethnic Albanian majority.
with AP
