The Turkish giants, who had not reached the last eight since 2001, progressed from an engrossing last 16 tie after winning 3-2 on the night in Germany.
Schalke had taken the lead through Roman Neustadter but their former player, Hamit Altintop, and tournament joint top goalscorer Burak Yilmaz hit back in the space of five minutes.
Michel Bastos then equalised in the second half to give Schalke hope of a dramatic comeback but Gala substitute Umut Bulut settled the tie in stoppage time.
From kick-off, it seemed only a matter of time until the Schalke goal was breached but it was the hosts who took a 17th minute lead when Neustadter took advantage of poor defending at a set-piece.
But Altintop brought the tie back to life in the 37th minute, when Didier Drogba squared a free-kick to the Turkey international 25 yards from goal and his fierce blast went in off the post.
And, five minutes later, Gala had taken the lead when Yilmaz shrugged off the challenge of Benedikt Howedes and lifted over Hildebrand for his eighth Champions League goal in as many games to go level with Cristiano Ronaldo in the scoring charts.
Jens Keller's Schalke started the second half in positive fashion and Bastos levelled in the 63rd minute after a bit of a scramble in the Gala box.
Schalke were then throwing everything forward for the goal which would take them through, however, the hosts left themselves wide open to the counter and Bulut made them pay in injury time.
January signing Drogba could not hide his pleasure at the victory after the final whistle and he told Sky Sports: "I am delighted with the way we played. We showed some character. Tonight, we came with a fantastic attitude."
Having secured the win, Galatasaray will join the likes of Barcelona and Real Madrid in Friday's quarter-final draw, and Drogba's fellow winter window signing, Wesley Sneijder, told Sky Sports: "We did a great job and now we have to see. We have to wait for the draw. It will be difficult."
|FC Schalke 04||Team Statistics||Galatasaray|
|1||1st Half Goals||2|
|12||Shots on Target||8|
|7||Shots off Target||6|
|46||1st Half Poss.||54|
|72.8||2nd Half Poss.||27.2|