The Freedom Party (FPOe) gained 25.7 percent of the votes, just ahead of the ruling conservative People's Party (OeVP) which stood at 24.7 percent, results published by Austrian media showed with almost all the votes counted.
The Social Democrats (SPOe) followed in third place at 23.2 percent, trailed by the Greens -- which currently rule Austria as junior partners of the conservatives -- at 10.7 percent, down from 14 percent in 2019.
The conservatives slumped from the almost 35 percent they had gained in the last EU elections.
Chancellor Karl Nehammer said he had heard voters' "message" and would seek to address their concerns ahead of national elections later this year, including cracking down on "illegal migration".
The anti-immigrant FPOe's share surged on the other hand -- it was up from 17 percent in 2019, when the party suffered from a string of corruption scandals.
Taken over by radical leader Herbert Kickl in 2021, its success is attributed above all due to his rebellious Covid anti-vaccination stance.
On Sunday, Kickl, 55, hailed "the history that the voters have written" to open "a new era in politics in Austria and Europe".
"This era is characterised by the fact that it is the people who are at the centre and not the disconnected elites of the system," he told cheering supporters in downtown Vienna, adding the "next step is the chancellery".
International media were barred from the party event, with the FPOe citing space restrictions.
The FPOe is also expected to top the vote in national elections expected to be held in September, but it remains to be seen if it can find partners to form a majority to govern.
The party -- founded in the 1950s by former Nazis -- has been part of a ruling coalition several times but has never governed.
During the pandemic, the party seized on anger over strict measures such as nationwide lockdowns and a mandatory vaccination law that was later scrapped.
On the war in Ukraine, Kickl has defended Austria's neutrality, criticising the EU sanctions against Moscow.
During the campaign, the party put up posters showing European Commission head Ursula von der Leyen hugging Ukrainian President Volodymyr Zelensky, urging voters to "stop the EU madness".
In total, 6.4 million people were eligible to vote in Austria, which has 20 seats in the 720-seat EU parliament. Seven parties fielded candidates.
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