Javier Milei landed some 56 percent of the vote versus just over 44 percent for his rival, Peronist Economy Minister Sergio Massa, who conceded.
"The model of decadence has come to an end, there's no going back," Milei said in a defiant speech after the result, while also acknowledging the challenges that face him.
"We have monumental problems ahead: inflation, lack of work, and poverty," he said. "The situation is critical and there is no place for tepid half-measures."
In downtown Buenos Aires hundreds of Milei supporters honked horns and chanted his popular refrain against the political elite - "out with all of them" - as rock music played from speakers. Some people set off fireworks as excitement spread.
"We came to celebrate this historic triumph," said Efrain Viveros, a 21-year-old student from the province of Salta. "I'm honestly ecstatic. Milei represents change, for the better. With Massa we'd have had no future, our future has returned."
Some Argentines had characterised the vote as a choice of the "lesser evil": fear of Milei's painful economic medicine versus anger at Massa and his Peronist party for an economic crisis that has left Argentina deeply in debt and unable to tap global credit markets.
Milei has been particularly popular among the young, who have grown up seeing their country lurch from one crisis to another.
The victory of Milei, a 53-year-old economist and former TV pundit, has broken the hegemony of the two leading political forces on the left and right - the Peronists that have dominated Argentine politics since the 1940s and its main opposition, the Together for Change conservative bloc.
"The election marks a profound rupture in the system of political representation in Argentina," said Julio Burdman, director of the consultancy Observatorio Electoral, ahead of the vote.
The campaign of Massa, 51, an experienced political wheeler-dealer, had sought to appeal to voter fears about Milei's volatile character and plans to cut back the size of the state.
"Milei's policies scare me," teacher Susana Martinez, 42, said on Sunday after she voted for Massa.
Milei is staunchly anti-abortion, favours looser gun laws and has criticised Argentine Pope Francis. He used to carry a chainsaw in a symbol of his planned cuts but shelved it in recent weeks to help boost his moderate image.
Who is Milei in Argentina?
Milei—53—an economist and former television personality with little political experience, burst onto the traditionally closed Argentine political scene with a brash style. During his election campaign, he floated so many proposals to upend a broken economy and government.