The People's Party (OVP), which Kurz has led since May, is widely expected to form an alliance with the Freedom Party (FPO), putting the far right in an Austrian governing coalition for the first time in more than 10 years.
According to preliminary results
, the OVP has gained the largest share of Parliament's powerful National Council, with more than 31% of the vote. That should translate to 61 of 183 seats.
Kurz gave a speech at his party headquarters in Vienna following early results, promising a new style of leadership.
"We have been given a great responsibility by the voters today. We should be aware of that and that people are putting a lot of hope in our movement. There is a lot to do. It's about establishing a new political style in our country," he said.
Detailed results will be announced October 19 and full official results will be declared on October 31.
Sunday's election was widely watched across Europe, where there has been a rise of populist far-right parties, many of which have campaigned on anti-immigration platforms.
Unlike Germany's far-right AfD party -- which won its first seats in the Bundestag in the federal elections there last month -- the FPO has a long history in Austria's Parliament and was part of a coalition government between 2000 and 2005.
The FPO came in second with 27.4% of the vote and is predicted to take 53 seats, according to preliminary results
. Current Chancellor Christian Kern's Social Democrats (SPO) were just behind on 26.7% and 52 seats.
While the result appears to be a clear win for the conservatives, it also puts the far-right FPO in the powerful position of kingmaker, after a campaign period that saw a marked shift to the right.
Austria rejected a far-right presidency less than a year ago. But that vote was a close one, with the FPO's Norbert Hofer missing out by a narrow margin.
Immigration takes center stage
Immigration has dominated the campaign. Kurz has taken a hard line, calling for limits on the number of refugees entering Europe
and benefits cuts for EU migrants living in Austria.
Austria was governed by a coalition led by Chancellor Kern's SPO and Kurz's OVP, but that partnership collapsed in May, prompting a snap vote.
The decision to call an early election also followed the resignation of OVP's former leader and Austria's Vice-Chancellor Reinhold Mitterlehner, who said the government was riven by infighting.
With immigration dominating the campaign, the social issues at the heart of the SPO platform -- including wealth redistribution and fighting unemployment -- have been largely ignored.