The sigh of relief that followed Alexander Van der Bellen’s victory in Austria’s rerun presidential election on Sunday could be heard all over Europe. After the twin traumas of Trump and Brexit, centrist parties, social democrats and liberals of all stripes had feared another triumph for the advancing forces of nativist populism represented by Van der Bellen’s rival, the far-right Freedom party’s Norbert Hofer.

Instead, Europe and its much-battered political incarnation, the European Union, have won a reprieve – although probably temporary. And Austria has escaped the odium of being the first modern-day democracy to pick as its head of state a political extremist whose party traces its ideological roots back to the strident neo-Nazism of its best-known leader, the late Jörg Haider.

Van der Bellen, a left-leaning, pro-Europe moderate backed by Austria’s Greens, was estimated to have won by an unexpectedly large margin of 7%. The initial contest in May gave him a 1% lead or less, an outcome that was challenged by Hofer.

The result will give a boost to likeminded politicians across Europe who also face potent electoral challenges from the far-right next year, notably in France. “What happens here today has relevance for all of Europe,” Van der Bellen said before casting his ballot, pointing to Hofer’s strong anti-EU, anti-immigrant, nationalist stance.

As the projected result was declared, Sigmar Gabriel, the German vice-chancellor who leads his country’s centre-left Social Democrats, expressed agreement and heartfelt relief. “A load has been taken off the mind of all of Europe,” Gabriel said. The result was “a clear victory for good sense against right-wing populism”.

Manfred Weber, leader of the largest centre-right grouping in the European Parliament, also read wider significance into the result. In effect speaking for Brussels, target of much nationalist ire, Weber said on Twitter: “Austrians send clear pro-European signal. The European right-wing populists’ party is off for now.”

Some even set the outcome in a global context – unusual treatment for Austria, a minor EU member whose politics are largely obscure. Ulrich Kelber, a German minister, suggested the vote reflected a backlash against Donald Trump’s victory in last month’s US presidential election. “Perhaps Trump’s election was the turning point. The liberal majority pushes back,” Kelber said. That remains to be seen.

The Austrian president’s functions are largely ceremonial, unlike in France, where a successor to François Hollande, the Socialist president, will be elected next spring. The contest will pit François Fillon, a centre-right Republican, against Marine Le Pen, leader of the far-right National Front. The French left is in disarray. Le Pen endorsed Hofer and sent him a message of support as voting began on Sunday.

Claims that his defeat represents the beginning of a liberal backlash will not worry her excessively. The FN appears confident that the feelings of alienation and disillusionment with the political establishment expressed by many French voters will translate into a strong if not decisive showing in the presidential poll.

Geert Wilders, the anti-immigrant, Islamophobic Eurosceptic who is currently riding a wave of support in the Netherlands, also backed Hofer. But he, too, is likely to play down the spillover effect of the Austrian result on the Dutch elections due in March.

By way of contrast, Germany, more closely related to Austria’s political culture and similarly troubled by issues of immigration and national identity, may see in Sunday’s outcome a sign of what to expect in next September’s federal elections, which will be contested by the populist Alternative for Germany.

Comforting though Van der Bellen’s victory may be for European progressives, the fact remains that slightly more than 46% of Austrian voters backed Hofer’s candidacy. That figure reveals a nation that is sharply divided, and overall, more rightwing and intolerant in its outlook than previously.

In this respect, similarities may be drawn with Britain, where the anti-immigrant Ukip has attracted increased support in recent years, and where 52% of voters opted in June to turn their backs on Europe and the EU.

Van der Bellen stressed the linkage with Brexit during his campaign. He predicted Hofer, if elected, would push for Austria to hold its own “Oexit” referendum. This prospect has been set back – but the idea will not go away entirely. Although the battle for the presidency is over, Austria’s larger, ongoing political struggle now enters its next stage.

With parliamentary elections due in 2018, the Freedom party is leading in opinion polls, with roughly one-third support. Van der Bellen has vowed to stop the formation of a Freedom party-led government, even if it were to win a general election. Such action could provoke a constitutional crisis.