The Al-Ahli center-back smashed home after just 58 seconds and produced a towering header on the hour mark to double Turkey's lead.
Michael Gregoritsch quickly pulled a goal back for Austria and only a stunning save from Mert Gunok denied Christoph Baumgartner a dramatic equalizer deep into stoppage-time.
Turkey advance to face the Netherlands in the last eight in Berlin on Saturday.
"I'm happy for him, for the team, for the country and I'm happy for the group we have created here," said Turkey boss Vincenzo Montella of Gunok's stunning stop.
"Well done to Mert. That's his job to make saves and we are glad he made a match-winning save in the final minutes."
Austria had emerged as dark horses to go far on the perceived weaker side of the draw after topping a group including France and the Netherlands.
Ralf Rangnick's side had also thrashed Turkey 6-1 in a friendly in March, but this time they failed to recover from a nightmare start.
"I cannot imagine we are going home. We thought we would continue our journey here and get ready for the next few games and the players feel the same," said Rangnick.
"That's how it works with knockout games, one team progresses and the other doesn't. Of course disappointment is the main sentiment."
Real Madrid's Arda Guler was a constant threat to the Austrian defense and his teasing delivery from a corner caused chaos inside the first minute.
Baumgartner's clearance off the line hit team-mate Stefan Posch, goalkeeper Patrick Pentz clawing it out to Demiral, who lashed into the roof of the net for the second fastest goal ever at a European Championship.
Turkey's raucous fans exploded in a deafening celebration, but they were nearly brought back down to earth straight away.
Baumgartner fired inches wide form the edge of the box before Demiral somehow prevented the RB Leipzig midfielder from scoring at his home ground as a dangerous Austrian corner flashed across goal.
Montella said that absent captain Hakan Calhanoglu, who was missing due to suspension, was "irreplaceable."
However, Guler assumed responsibility in the playmaking role and nearly produced a stunning second for his side with an audacious attempt from the halfway line that drifted wide.
Gunok to the rescue
Rangnick made two changes at half-time, including the introduction of Gregoritsch, who scored a hat-trick when the sides met three months ago.
He initially got the reaction he desired as Austria began cutting through the Turkish defense.
Gunok spread himself brilliantly to deny Marko Arnautovic when one-on-one before Bayern Munich's Konrad Laimer lacked the finish to a fine run through the middle.
However, they failed to learn their lesson from Guler corners as another inviting delivery was powered home by Demiral.
In doing so the 26-year-old became the first European defender to score twice in the knockout stages of a major tournament since Lillian Thuram for France at the 1998 World Cup.
It was also a set-piece at the other end that got Austria back in the game seven minutes later.
Posch flicked on Marcel Sabitzer's corner for Gregoritsch to sweep high into the net.
A torrential downpour could not drown out a sensational atmosphere in Leipzig, with Turkey's huge expat population in Germany again making it like a home game for Montella's men.
The fervent atmosphere turned sour as Sabitzer was struck by an object thrown from the crowd as he prepared to take a corner.
Baris Alper Yilmaz was denied a third for Turkey by a fine save from Pentz in stoppage-time.
That nearly proved crucial as Gunok then had to produce a remarkable stop to prevent Baumgartner's downward header finding the far corner.
It was an incredible save reminiscent of England keeper Gordon Banks' iconic stop from Pele in the 1970 World Cup.
"It is difficult when you have Banks in the goal," added Rangnick.
But Turkey stood firm amid an aerial bombardment from Austria to reach the quarter-finals of a major tournament for the first time since Euro 2008.