The home side took the lead in the fifth minute when Rurik Gislason's flick-on was turned in by former Bolton Wanderers midfielder Braaten.
There was controversy 15 minutes later as Galatasaray thought they had equalised through Bruma, whose strike was contentiously ruled out for offside.
End-to-end action followed, with Braaten missing a golden opportunity to put his side 2-0 up before the break and Didier Drogba putting a volley into the stand at the other.
Drogba had another chance to draw his side level midway into the second period, the former Chelsea man with an instinctive effort from inside the box that was turned behind by Johan Wiland.
The game turned scrappy in the final stages as Galatasaray struggled to get any balls up to Drogba in the attack, allowing Copenhagen to hold on for the win.