With Galatasaray needing to score thanks to Schalke away goal three weeks ago, they came out of the blocks the quicker of the two teams, with Hamit and Didier 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, S04 had offered very little. Bastos made Fernando Muslera work with a tame effort and Joel Matip headed over from a corner, but, in truth, their opener was against the run of play.
Benedikt Howedes beat Drogba to another Jefferson Farfan in-swinger, Semih Kaya failed to clear the loose ball and Matip forced it into the path of Neustadter, who powered an effort into the back of the net.
The onus was now on Schalke to take control of the game truly, but Galatasaray responded to their deficit well. Wesley Sneijder fired a superb first-time strike wide of the post, before they found their equaliser at long last.
Selcuk Inan declined the opportunity to swing a free kick into the box, instead squaring it to Hamit, who shot from all of 35 yards, his effort deflecting in via the inside of the post.
Four mere minutes later, and they were ahead. Albert Riera beat Marco Hoger to a 50-50 ball and his clearance fell to Burak, who beat Howedes in a foot-race and lifted the ball over the onrushing Hildebrand.
There was still time for Bastos to force Muslera into a rare save in first-half stoppage time, but in truth, Schalke had not done anywhere near enough to trouble the Uruguayan.
And, in the immediate aftermath of the restart, things did not improve for the hosts, but Galatasaray had nothing to show for their territorial advantage but a Burak effort which Hildebrand saved.
At the other end, 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 which the former Lazio keeper did well to parry.
Gradually, they began to creat more and more chances. Some lovely quick feet from Draxler left Felipe Melo and Semih wonderstruck, but the youngster could only shoot wide.
The next attack, though, yielded the equaliser. Marco Hoger cracked a strike against the woodwork, and in the ensuing melee in the Gala box, Atsuto Uchida steered the ball to Bastos, who applied the finish.
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 and Amrabat secure his side's place in the last eight with the last kick of the game.