He however refused to rule out a legal challenge to the result of Sunday's election, saying his party was evaluating multiple claims of irregularities made by supporters on social media.
"A new political era has already dawned in Austria. We have already written history several times, we are stronger than ever," Strache said in Vienna.
"We have scored a success against the whole creaking system."
On Monday ecologist Alexander Van der Bellen was declared winner of the election to the largely ceremonial but coveted presidency, beating the FPÖ's Norbert Hofer by 50.3 percent to 49.7 percent -- a difference of just 31,026 votes.
Strache, 46, told a news conference that "anyone interpreting this result, like in the media yesterday... as a defeat, then I say they have simply not properly understood this election."
Mirroring the rise of other populists in Europe and beyond, Hofer tapped into unease about immigration and Austria's faltering economy to win support not just among poorer, less educated voters but across the board.
Hofer, 45, is presented as the friendly, moderate face of the FPÖ but who nevertheless believes Islam is not part of Austria, opposes gay marriage and says that if Austria were not a member of the EU, he would vote against it joining.
On Tuesday he said that his strong showing in the election showed that the FPÖ, formed by former Nazis in the 1950s and led in the 1990s by the controversial Joerg Haider, was anything but "extreme-right".
"An extreme-right party would not score more than two percent. That's the number of stupid people," Hofer said.
His narrow defeat was greeted with relief by mainstream politicians across Europe, but other far-right groups like France's National Front and Italy's Northern League said Hofer's strong showing was a sign of things to come.
The FPÖ is now looking ahead to the next general elections, due in 2018. The party is leading the opinion polls with more than 30 percent while the two centrist parties in the governing coalition look set to fall short of a majority.
"I will do everything in my power to make sure that the FPÖ wins more than a third of the vote at the next parliamentary election. It will be impossible to keep us out of government," Hofer said on Tuesday.
Neither of the two main parties, which have dominated Austrian politics since 1945, even made it into the runoff for the presidency, scoring a dismal 11 percent each and prompting chancellor Werner Faymann to resign on May 9.
"Austria has a particularly concentrated version of a problem that you see elsewhere in Europe," Heather Grabbe, fellow at the European University Institute in Florence, told AFP.
"People are getting fed up with the mainstream parties and feel they have run out of ideas."