Galvanised by a superb swerving strike by Hamit Altintop against his old club, the Turkish side dominated the first half, held on for grim life in the second and snatched the last 16 second leg 3-2 with a stoppage-time goal on the break against bewildered opponents.
Their 4-3 aggregate win justified their decision to splash out on big-name signings Wesley Sneijder and Drogba in January while Schalke became the first German side to exit the Champions League proper this term.
With Borussia Dortmund already through and Bayern Munich set to clinch a last-eight place on Wednesday when they host Arsenal with a 3-1 lead in the bag, Germany's three teams had been in imperious form, losing only one match between them before Tuesday.
"It's incredibly disappointing for us," said team captain Benedikt Howedes, whose team are fourth in the Bundesliga and face a battle to qualify for the Champions League next season.
Ivory Coast forward Drogba, who missed a penalty in a league defeat against Genclerbirligi on Friday, was hugely influential on his favourite Champions League stage after his short stint in Chinese football.
The 35-year-old triumphed in last season's competition with Chelsea after scoring a late equaliser in the final against Bayern and netting the winning spot-kick in the shootout.
He looks hungry for more success but Galatasaray's progression to the quarters for the first time since 2001 seemed unlikely when Roman Neustadter gave Schalke a 17th-minute lead completely against the run of play.
It was wiped out by Altintop's effort in the 37th minute when he collected a free-kick and scored with a 30-metre shot which swerved away from Timo Hildebrand and went in off the foot of the post.
It was Altintop's first goal since he joined Galatasaray from Real Madrid during the close season and came against the club where he spent four seasons.
Burak Yilmaz then put Galatasaray ahead with a breakaway goal five minutes later, making him the competition's joint top scorer with Cristiano Ronaldo on eight goals.
"We completely went to sleep in the first half," said Schalke 04 coach Jens Keller, whose team had been boosted by a 2-1 win over bitter rivals Borussia Dortmund on Saturday. "We didn't get into the game."
The second half was completely different as Galatasaray sat on their lead and Schalke poured forward.
Schalke's dominance was rewarded when Brazilian Michel Bastos levelled just after the hour but, despite continuing to press forward, they could not find a winner and instead lost to Umut Bulut's 94th minute breakaway effort.
Drogba's contribution began when he nearly scored from outside the area in the fifth minute and then saw a fierce left-foot effort palmed away by Hildebrand one minute later.
Sneijder, signed from Inter Milan, was less impressive although the Dutch playmaker nearly scored with a first-time side-footed effort from the edge of the box.
"We definitely deserved it. We made a lot of mistakes in the second half and dropped back and they made it difficult," Sneijder told reporters.
"We wanted to go for the third, but we didn't do it and they kept pushing [until we scored a third late on]. I think Schalke played well but with all our team spirit, it's a deserved victory."
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