The German side mounted a brave second-half effort, but was ultimately unable to overcome the Turks, who claimed a hard-fought away win to stay alive.
Roman Neustadter put the hosts ahead in the 17th minute, but goals from Hamit Altintop and Burak Yilmaz in a five-minute span late in the first half turned the game on its head. Michel Bastos equalized after the break, but a third goal proved to be a step to far for Jens Keller's side, and Nordin Amrabat ended the contest in second-half stoppage time.
With Galatasaray needing to score thanks to Schalke' saway goal three weeks ago, the Turks came out of the blocks the quicker of the two teams, with Atintop and Drogba firing wide in the early minutes. Timo Hildebrand was then forced into two good saves in quick succession, denying the Ivorian icon after some superb skill and palming wide from Selcuk Inan, who had been sent clear.
At the other end, meanwhile, Schalke had offered very little. Bastos made goalkeeper Fernando Muslera work with a tame effort, and Joel Matip headed over from a corner, but, in truth, the club's opener was against the run of play.
Benedikt Howedes beat Drogba to a Jefferson Farfan in-swinging corner kick, and after Semih Kaya failed to clear the loose ball, Matip forced it into the path of Neustadter, who powered an effort into the back of the net.
The onus was on Schalke to take control of the game truly, but Galatasaray responded to the deficit well. Wesley Sneijder fired a superb first-time strike wide of the post before the Turks found their equalizer.
Inan declined the opportunity to swing a free kick into the box, instead squaring it to Altintop, who shot from 35 yards, his effort deflecting in via the inside of the post to torch his former team in his home town.
A mere five minutes later, Galatasaray turned the tie on its head. Albert Riera beat Marco Hoger to a 50-50 ball and his clearance fell to Yilmaz, who beat Howedes in a foot race and lifted the ball over the onrushing Hildebrand.
Schalke rallied hard in the second half, and Julian Draxler had an effort charged down by Semih, while Muslera was forced into a rare save when Teemu Pukki cut onto his right foot and sent a low effort towards the corner that the former Lazio keeper did well to parry.
Soon enough, the hosts had equalizer. Marco Hoger cracked a strike against the woodwork, and in the ensuing melee in the Galatasaray box, Atsuto Uchida steered the ball to Bastos, who applied the finish.
Schalke still needed another to combat Galatasaray's away-goal edge, and Draxler had another great effort denied by Muslera, before Jefferson Farfan's strike appeared to hit a Galatasaray arm. Howedes sent a free header at the near post over, and Muslera got off his line brilliantly to smother a Draxler cross at close range.
Bastos sent a further effort into the keeper's arms, but in the end, Galatasaray broke with Schalke having numbers forward, and Amrabat secured his side's place in the last eight with the stoppage-time effort.