With the contrasting domestic form of the two sides, Rangers needed to start well and unsettle their high-flying opponents, and the decision of interim manager Barry Ferguson to switch to a three-at-the-back formation did just that.
With only six minutes played, Cyriel Dessers capitalised on some poor defending to take the ball around goalkeeper Irfan Can Egribayat, with his shot just creeping over the goal line before Caglar Soyuncu could clear, giving the visitors a surprise early lead.
The hosts mounted something of a response to this early setback, with Youssef En-Nesyri and Edin Dzeko both coming close before substitute Alexander Djiku showed the finishing instincts of a striker, sticking his right leg out to hook a flicked ball from a corner goalwards beyond Jack Butland.
Rangers were enduring a tricky spell, however, with three minutes remaining in the first half, they re-established their lead.
After winning possession back in midfield, Rangers were on the offensive, with Vaclav Cerny freeing Dessers inside the box, allowing the Belgian to play the ball back across the area to the Czech forward, who rolled it past Egribayat, stunning the home fans once more.
The 'Gers would have expected a second-half onslaught from Sari Kanaryalar, however, they could easily have found themselves in a more commanding position were it not for some poor fortune for Dessers.
The striker thought he had found the net for a second time six minutes after the restart when he followed up on Nicolas Raskin’s saved effort and fired home, only for VAR to intervene with an offside call.
Then, just six minutes later, the forward beat Egribayat at his near post with a precise effort, but it was a case of déjà vu for the Belgian as the VAR again adjudged him to have been offside.
With both sides tiring, the game descended into a series of counter-attacks that made for thrilling viewing.
And having spurned numerous opportunities to draw level, the home side were punished with 10 minutes remaining when Cerny raced upfield and slotted past Egribayat, sealing a famous away win.
Rangers have now lost only one of their last eight Europa League away fixtures and will be hot favourites to progress to the quarter-finals when the tie concludes at Ibrox next week.
Flashscore Man of the Match: Vaclav Cerny (Rangers)