Demiral opened the scoring after 57 seconds for the second-fastest goal in Euros history - and the quickest ever in the knockout stage - and nodded in the second in the 59th minute.
"Our team spirit showed until the last minute," Turkey coach Vincenzo Montella said.
In a tournament of quickfire goals, Albania had scored after 23 seconds against Italy in the group stage.
Both of Demiral's strikes followed corner kicks from young star Ardar Guler.
Substitute Michael Gregoritsch pulled one back in the 65th minute but Austria failed to level despite sustained pressure as their strong run came to an abrupt end.
A sensational last-gasp save from goalkeeper Mert Gunok denied Christoph Baumgartner as Turkey held on to earn their place in the last eight.
Austria's goalscorer Gregoritsch described Gunok's stop as "one of the best saves I have ever seen".
"I'm sure that if Baumi had scored to take the game to extra time, we would have gone on to win the match," Austria coach Ralf Rangnick said.
"The only thing we could be accused of is that we didn't score enough goals from the chances we had, and that we didn't work well twice for set pieces."
Austria had impressed in the group stage with victories over Poland and the Netherlands, and had thrashed Turkey 6-1 in a March friendly, but all that counted for nothing on a rainy night in Leipzig.
Boosted by their large and raucous fan base, Turkey will next face the Netherlands on Saturday in Berlin - a side Montella said he expected to "fight like lions".
"We will gather our mental and physical strength and keep following our path," he said.
They will be without Ismail Yuksek and Orkun Kokcu, who will both be suspended after being booked on Tuesday.
- with AP
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