By FOX Sports staff
Heading into the second leg at the Veltins-Arena with the tie evenly poised at 1-1, Schalke drew first blood when Roman Neustadter pounced on a loose ball to open the scoring in the 17th minute.
However, the Turkish Super Lig giants equalised eight minutes before the break through a blistering Hamit Altintop strike, before taking the lead five minutes later when Burak Yilmaz raced through to score.
Still, Schalke refused to be deterred and made it 2-2 in the 63rd minute when Michel Bastos fired home from close range, but they were unable to find a vital third goal that would have seen them through, and the visitors went on to wrap up the win in injury-time when substitute Umut Bulut netted on the rebound.
Here, FOX Sports breaks down where Tuesday's game was won and lost.
One player that was really exposed by Galatasaray was Royal Blues captain Benedikt Howedes, who was caught in possession on several occasions and regularly lost track of his direct opponent.
Had it not been for Joel Matip, who bailed his centre-back partner out several times with vital interceptions, the hosts could have lost by an even bigger margin.
Right-back Atsuto Uchida however was a real bright spark for the Bundesliga side as he defended well against Selcuk Inan and even pushed forward to good effect, showing great composure to lay on an assist for Bastos
Hildebrand - 7 | Uchida - 7.5 | Howedes - 4 | Matip - 7 | Kolasinac - 5
Schalke coach Jens Keller's decision to deploy two defensive players - Neustadter and Marco Hoger - in the middle of the park backfired as his side were sorely lacking in creativity, especially in the first half.
Some might have expected that to come from the wings and while Bastos tried his best all game long, the same cannot be said for Farfan, who failed to get the better of his direct opponent and was woeful with his deliveries into the box, most of which were easily cleared by the visiting defenders.
Farfan - 5 | Hoger (off 85') - 6.5 | Neustadter (off 46') - 7 | Bastos - 7.5
The Schalke attack certainly took awhile to get going as they appeared toothless for most of the first half, but after the break, Julian Draxler and Teemu Pukki showed why they are both regarded highly at the club.
Pukki started making his way into dangerous positions and was never afraid to test Fernando Muslera, but it was Draxler who really caught the eyes with his pace and skill on the ball.
Still only 19, the Germany international - deployed as a support striker - nearly singlehandedly dragged his side across the finish line, and is certainly one for the future.
Draxler - 8.5 | Pukki (on 85') - 7
Keller made a bold move to replace goalscorer Neustadter at the break, but the substitution failed to have the desired impact as Christian Fuchs struggled to pose any sort of threat down the left, despite his reputation as a rampaging full-back.
The Schalke boss also took too long to make his next move as he brought Chinedu Obasi and Max Meyer on with five minutes remaining; both players having little time to make a difference.
Fuchs (on 46') - 6.5 | Obasi (on 85') - N/A | Meyer (on 85') - N/A
Overall, the Galatasaray defence worked well as a unit on Tuesday evening - a key reason why it was them who eventually progressed to the last eight.
Although he conceded two goals, Muslera made a series of fine stops to keep Draxler and Pukki at bay, and he was also ably supported by Semih Kaya, Dany Nounkeu and Albert Riera.
Nonetheless, their good work was almost ruined by a certain Emmanuel Eboue, the one-time Arsenal defender who was regularly caught out of position. His failure to pick up Bastos also allowed the Schalke man to level the scores at 2-2 shortly after the hour mark.
Muslera - 7.5 | Eboue - 6.5 | Semih Kaya (off 79') - 7 | Nounkeu - 7.5 | Riera - 7
While he was much maligned during his spell at Juventus, Felipe Melo has shown why the Serie A side bought him in the first place since moving to the Turk Telekom Arena on loan. The Brazil international provided a strong shield in front of his back four and was also tidy with his distribution, something that can't always be said about him.
On the right, Altintop was a real threat for the visitors and equalised with a terrific strike, while Selcuk Inan provided a constant supply of crosses on the other flank.
Wesley Sneijder however was extremely disappointing as he struggled to have any influence on the game, apart from a snapshot that sailed just wide of Timo Hildebrand's post, and a visionary ball that nearly send a team-mate through on goal early in the second half.
Melo - 8 | Hamit Altintop - 8 | Selcuk Inan - 7.5 | Sneijder (off 70') - 5
Just like he was at Chelsea, Didier Drogba is still prone to going down under a challenge a bit too easily, but he also remains brilliant at holding up the ball in the attacking third and retained possession well for Galatasaray, even though he didn't have many shooting opportunities.
His strike partner, Burak Yilmaz, had a torrid start to the game as he struggled to perform the basic task of controlling the ball, but his predatory instincts came to the fore just before the break when he chased down a clearance from Riera and burst through, before expertly lifting the ball over Hildebrand.
In doing so, the 27-year-old equalled Marouane Chamakh's record of scoring in six consecutive Champions League games, and with Galatasaray through to the quarters; he could just better that in a few weeks' time.
Drogba - 7 | Burak Yilmaz (on 86') - 7.5
Having seen Sneijder fail to get into the game, Gala coach Fatih Terim decided to replace him with Nordin Amrabat in the 70th minute but the ex-PSV Eindhoven man failed to fare any better.
Still, one substitution that did bear fruit was that of Umut Bulut, who wrapped up the win just eight minutes after coming on when he raced through and beat Hildebrand at the second time of asking, after his first shot had been blocked by the goalkeeper's feet.
Amrabat (on 70') - 6 | Gokhan Zan (on 79') - N/A | Umut Bulut (on 86') - 7