Hofer's announcement puts Alexander Van der Bellen in the clear to assume the presidency. Official results have not yet been announced, but exit polls from state broadcaster ORF showed Van der Bellen with 53.6% of the vote.
While the Austrian president's role is largely ceremonial, the vote was closely watched around the world as a gauge of wider anti-establishment sentiment
following the election of US President-elect Donald Trump
and Britain's vote in June to leave the European Union.
"I am incredibly sad it didn't work out," Hofer wrote in a concession statement on Facebook.
"I would have loved to look after Austria. I congratulate Alexander Van der Bellen to his success and ask all Austrians to stick together," he added. "We are all Austrians, no matter what we decided today. Long live our home Austria."
At a press conference shortly after, Hofer told CNN he was "very sad" about the result, adding "he wouldn't be human if he wasn't disappointed."
He hugged his youngest daughter, who appeared to have been crying, and said after the election he planned on "kicking back, relaxing with my wife and reading 'The Old Man and the Sea.'"
Austria votes for 'freedom, equality, solidarity'
Speaking on Austrian state TV, Van der Bellen credited his apparent victory to "the engagement of hundreds of thousands" of people.
"That's how a big movement developed. I have always campaigned for a pro-European Austria. This is about values: freedom, equality and solidarity. Also to include those that don't do that well in the current economic system," he said.
"I will actively speak to all voters, including those of Hofer's party. They have real concerns that have to be taken seriously by the president."
Green Party's significant gains
Van der Bellen's apparent win marks a significant change to Austria's political landscape, which has been dominated by two centrist parties since the end of the war.
The original presidential vote was held in May, with Van der Bellen, a former Green Party leader, defeating Hofer by little more than 30,000 votes in a tightly fought contest decided by mail-in ballots.
But the results were challenged by Hofer and his party, and were eventually annulled over concerns about how some ballots were handled
This time around, Van der Bellen's margin was expected to have increased tenfold, to more than 300,000 votes, he said in an address on Austrian state TV.
"It was not a rerun but a new election," Van der Bellen said, adding that voters had been influenced by the changing political landscape over the last six months.
"The world around us has changed. We had the Brexit referendum in Britain and the election of Donald Trump in America," he said.
Van der Bellen said the result sent a "red, white, red signal (referring to the colors of the Austrian flag) of hope" to other European countries.
A win for Hofer would have made Austria the first nation with a far-right head of state in Western Europe since the end of World War II.
The leader of France's far-right National Front, Marine Le Pen, congratulated Hofer's Freedom Party for fighting "with courage."
"The next legislative elections will be their victory," she said on Twitter.
Meanwhile, German Foreign Minister Frank-Walter Steinmeier said in a Twitter post that the "whole of Europe has heaved a sigh of relief" at the outcome.
French President Francois Hollande also congratulated Van der Bellen in a statement, saying "The Austrian people have chosen Europe, and they have chosen to be open."
European Parliament President Martin Schulz tweeted that Van der Bellen's victory was "a heavy defeat of nationalism and anti-European, backward-looking populism."
Van der Bellen, a 72-year-old economist whose parents spent time in a refugee camp before settling in Austria, has championed liberal migration policies.
He is an outspoken supporter of gay marriage, and despite his age, he has young fans who organized dance raves in his support.
In contrast, Hofer's party had risen on a populist groundswell sweeping the continent caused by the European Union's failure to make progress in the ongoing economic and migrant crises.
On Sunday, after casting his ballot, Van der Bellen said the Austrian election was "of significance for all of Europe."
"Outside of Austria, the election is perceived as something that does not only concern us Austrians," he said.
Following his victory in May,
which was later annulled, Van der Bellen told CNN's Christiane Amanpour that Europe "shouldn't dramatize" the growing popularity of far-right parties in Europe.
There were concerns in Brussels that Austria was on track to become the next "euroskeptic" state, with Hofer indicating support early in his campaign for a referendum, like Britain's "Brexit" vote, on the country's membership in the EU.
But Hofer softened his position on the EU, saying he preferred to stay in a reformed decentralized union.