Kurz heads the conservative People's Party and is on course to win the polls.
According to reports, the People's Party is expected to win 31 percent of the votes.
However, even with that percentage, the People's Party will finish short of the required number and is expected to form alliance with anti-immigration Freedom Party.
Following the initial trend, an overwhelmed Kurz told his supporters, "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," he said.
Before the polls, Kurz was Europe's youngest-ever foreign minister.
He was appointed in 2013, aged 27.
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