VIENNA — Supporters see him as the future of a country that tends to define itself more through its past than its prospects. Detractors consider him a presumptuous climber who out-demagogued populists in a ruthless grab at power.

Whatever their take, few Austrians dispute that at 31, their foreign minister, Sebastian Kurz, has already left an indelible mark on their central European homeland. After an intense campaign, he won a snap election on Sunday, setting him on track to become prime minister of Austria and one of the world’s youngest leaders.

“Thank you, thank you, thank you,” a visibly overwhelmed Mr. Kurz told cheering supporters who had packed into the ballroom of Vienna’s Kurhaus music hall on Sunday night. Roughly an hour earlier, initial returns had shown his center-right People’s Party, a Christian Democratic party founded at the end of World War II, to be in the lead, and Mr. Kurz did not want to miss the moment.

“Five months ago, we started with the goal to open the People’s Party and make it into a broad movement,” said Mr. Kurz, who dropped by the election party before heading off to previously scheduled television appearances. “We achieved that goal.”