Early on it was Gala who started better and Didier Drogba missed a pair of chances inside the first 10 minutes before Schalke struck from a corner.
Wesley Sneijder came within inches of an equaliser with a first-time shot and soon the Turkish side were level when former Schalke midfielder Hamit Altintop collected a quick free kick and swerved a 30-yard effort in off the post.
The turnaround was complete just four minutes later when prolific frontman Burak Yilmaz scored in his sixth straight European game by running on to a fortunate ricochet and beating Howedes for pace before toeing a lob over Timo Hildebrand.
Yilmaz came close to yet another goal eight minutes into the second half but Hildebrand's left hand was just strong enough to keep out the Turk's stinging effort.
Schalke went up the other end and Teemu Pukki saw his low shot parried away by Muslera as the Germans piled on the pressure knowing they needed two more goals to go through.
Howedes headed a chance wide for the hosts before heavy late pressure didn't pay off and Galatasaray ended up scoring on the counter to seal the win.
Nordin Amrabat streaked clear with all the blue shirts behind him and despite his first effort being kept out by Hildebrand, the ball fell kindly and he rolled it into the empty net.