"It is clear to me that Austria must play an important role within the European Union and when we are pro-European, we should not only stand for Europe but be an especially active participant in Europe," the 31-year-old conservative "whizz-kid" said.
"This is clearly what I will do in the coming years," he added.
Kurz's People's Party (ÖVP) won 31.5 percent of the vote on Sunday, near-complete results show, and his most likely coalition partner is seen as the populist Freedom Party (FPÖ), which came in third on 26.0 percent.
The FPÖ's manifesto says that Austria sovereignty and the "ideal of a Europe of fatherlands" are "increasingly under threat," particularly with the "aim of the EU being a centralised bureaucratic monster".
It wants EU sanctions on Russia lifted and for Austria to potentially join the Visegrad group of eastern and central European countries that has become a thorn in Brussels' side.
Kurz said he would visit Brussels on Thursday, saying he "deliberately" chose the EU capital for his first foreign trip after the election. He is due to meet Jean-Claude Juncker and Donald Tusk, the respective presidents of the European Commission and the European Council.
Juncker on Monday had pointedly wished Kurz success forming a "pro-European government" in a letter posted on Twitter, while German Chancellor Angela Merkel commented on the FPÖ result, calling it "a big challenge".
Kurz could be given the go-ahead to form a new government as early as Friday.