Mr Kurz is on course to become the world's youngest national leader.
The People's Party was set to win more than 31%. It is so far unclear whether the Social Democrats or the far-right Freedom Party will finish second.
Short of a majority, Mr Kurz's party could seek an alliance with the anti-immigration Freedom Party.
Addressing his supporters, Mr Kurz said: "It is time for change in this country. Today is a strong order for us, to change this country, and I say thank you to you all who made this possible.
"I'm overwhelmed, I am happy, and I look forward to working for Austria."Who is Sebastian Kurz?
Before the election, Mr Kurz served as Europe's youngest-ever foreign minister, after he was appointed in 2013 aged just 27.
In May 2017 he became the leader of the People's Party. He began his political career in the youth wing of the party, which he chaired before moving on to serve on Vienna's city council.
Nicknamed "Wunderwuzzi" (roughly translated - someone who can walk on water), he has been compared to the young leaders of France and Canada, Emmanuel Macron and Justin Trudeau.
Much like Mr Macron, Mr Kurz has created a movement around himself, rebranding the People's Party - which has been in power for more than 30 years - as "The New People's Party".What were the main issues?
Immigration was the dominant issue in the run-up to the vote, and Mr Kurz moved his party to the right in the wake of Europe's 2015 refugee crisis.
He appealed to conservative and right-wing voters with pledges to shut down migrant routes to Europe, cap benefit payments to refugees, and bar immigrants from receiving benefits until they have lived in Austria for five years.