After a 1-1 draw in the first leg, Roman Neustadter prodded Schalke into an early lead from a corner.
But Galatasaray hit back with two goals in five minutes
- a 25-yard strike from ex-Schalke midfielder Hamit Altintop and a cool
finish from Burak Yilmaz.
Michel Bastos levelled in the second half to give Schalke hope before Umut Bulut's late winner on the break.
The result means Galatasaray have reached the last
eight for the first time in 12 years and are the first Turkish side to
do so in five years.
Galatasaray included former Champions League winners Wesley Sneijder and Didier Drogba in their starting line-up.
Indeed, Drogba's last Champions League goal came in Germany from the spot in a penalty shootout as
Chelsea clinched victory
in last year's final against Bayern Munich.
And the Ivorian started in menacing form with two
quick-fire shots - the second beaten away by Schalke goalkeeper Timo
Hildebrand.
But it was the home side who took a 17-minute lead
against the run of play when Galatasaray failed to clear a corner and
Neustadter reacted first to steer the ball into the net.
The visitors were level 20 minutes later when Selcuk
Inan rolled a short free-kick to Altintop and he hit a fierce dipping
drive from 25 yards that crashed in off the inside of the post.
And Fatih Terim's side were ahead when Albert Riera pinched the ball in his own half and quickly released Yilmaz.
The powerful striker shrugged off the challenge of
Benedikt Howedes before cleverly lifting the ball over Hildebrand for
his eighth Champions League goal in eight games.
The goal silenced the normally buoyant Gelsenkirchen
crowd but, despite needing two goals to progress, Jens Keller's side
looked determined to give it their all in the second half.
Marco Hoger cracked the crossbar from the edge of the box moments before Bastos levelled in the 63rd minute.
Galatasaray failed to clear the ball from their box and
Japanese international Atsuto Uchida kept his cool to square for the
Brazilian to finish into an empty net.
Julian Draxler and Teemu Pukki had half chances to pinch a winner for Schalke but the Turkish defence held firm.
And they snatched a winner of their own on the night in
injury time when substitute Bulut beat the offside trap before putting
the ball past Hildebrand at the second attempt.
