Having topped Group D ahead of France and Netherlands, Austria were fancied to overcome Turkey in Tuesday's encounter in Leipzig.
However, it was Vincenzo Montella's side who would progress through to the quarter-finals, helped by a goal from Merih Demiral inside the opening 60 seconds.
The defender would double Turkey's lead before the hour mark, only for Michael Gregoritsch to halve the deficit with 24 minutes remaining.
Turkey would ultimately see out the contest to set up a showdown with Netherlands on Saturday evening, the winner of that tie going on to face Switzerland or England.
Rangnick heaps praise on Gunok save
While both teams can claim to have squandered opportunities throughout an entertaining fixture, it took a wonder save from goalkeeper Mert Gunok to send Turkey through to the last eight.
Christoph Baumgartner's free header at the back post appeared to be destined for the far corner, only for Gunok to miraculously claw the ball away for a corner.
Speaking after the match, Rangnick was left deflated by the intervention of the Besiktas stopper, comparing it to the famous save made by England's Gordon Banks from Pele at the 1970 World Cup.
He said: "We had time to score an equaliser, but it's difficult when they have Gordon Banks in goal! We didn't have the necessary luck and I believe if the game had gone to extra time we would have won.
Rangnick added: "I cannot believe we are going home today, we thought we were going to continue our journey. The four games we played were very entertaining, including today.
"Not everything went very accurately, but they were four very entertaining and intense games."
Rare defeat for Austria
Losing to Turkey represented just the third defeat that Austria have suffered in their last 20 matches, the other setbacks coming against Belgium and France.
More frustratingly for The Team, they thrashed their most recent opponents by a 6-1 scoreline in a friendly back in March.