In what might be interpreted as a bad omen by those fearing for the EU’s future, the large pole-mounted flag collapsed and hit the ground just inches from where Mr Strache was standing.
The vice chancellor, who is also the leader of Austria’s far-right Freedom Party (FPO), joked afterwards: “Today I saved and intercepted the EU.”
He helped raise the bloc’s flag again before walking off smiling.
The FPO formed a coalition with the conservative People’s Party (OVP) after elections in Austria in December last year.
The country became the only nation in western Europe to have a far-right presence in government, in a move that would have rattled EU leaders committed to open borders and liberal values.
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The FPO surprised people by showing support for the bloc in contradiction of the far-right’s longstanding rejection of it.
Austria’s President Alexander Van der Bellen only approved the coalition deal after he was assured the government would be pro-European.
He said: “In these talks among other things we agreed it is in the national interest of Austria to remain at the centre of a strong European Union and to actively participate in the future development of the European Union.”
The People Party’s 31-year-old foreign minister Sebastian Kurz became the country’s chancellor.
In the run-up to voting, both parties took a hardline on immigration, called for cuts to benefits for refugees in Austria, a crackdown on Islamic groups, and halting “illegal” immigration.
The FPO had earlier been in a coalition government in Austria between 2000 and 2005.
There was uproar among EU leaders at the time.
In September last year a group remembering Nazi camp victims published a list of 60 alleged antisemitic and racist incidents carried out by FPO members since 2013.
Analyst Alexandra Siegl said: “If they really changed their ideology, it is a question they can only answer themselves.
“I would say they changed their tactics and strategies mainly.”
(Additional reporting by Monika Pallenberg.)