Despite Austria’s status as the fast-starting side at Euro 2024, it was Türkiye who took the early lead, as Arda Guler’s corner was stabbed home by Demiral with barely a minute on the clock.
And as the unlikeliest of heroes, Demiral doubled his side’s lead before the hour mark in similar fashion, as Austria couldn’t contain Türkiye at the set piece.
Corners appeared to be the only opportunity for either side to score and saw Austria hit back through substitute Michael Gregoritsch.
An outswinger from Marcel Sabitzer was stabbed towards the far post, where an unmarked Gregoritsch poked past goalkeeper Mert Gunok to notch the 100th goal of the tournament.
In the final 10 minutes, as rain fell over Leipzig, Austria couldn’t find an equaliser, even as they piled men into the Turkish penalty area.
And conversely, Türkiye couldn’t add to their advantage, with more than one chance on the counterattack, notably to striker Baris Alper Yilmaz, who was denied by Patrick Pentz in the Austrian goal.
Austria had one final chance in the last seconds of added time, when Christoph Baumgartner’s diving header was kept out by Gunok, to all but send Türkiye into the last eight.
Elsewhere, Spain will meet hosts Germany in their quarter-final, and Portugal face France, both to be played on Saturday morning.
England will meet Switzerland in their quarter-final on Sunday, played in the hours before the Netherlands face Türkiye.