Braaten got the hosts off to a flying start, applying a glancing touch to Rurik Gislason's precise centre to turn the ball home six minutes into the Group B match.
Galatasaray clawed their way back into the game, hitting a post and having a goal disallowed for offside, but despite going close several times in a pulsating second half they could not fashion an equaliser.
Copenhagen are now third, a point ahead of Juventus and level on four with second-placed Galatasaray who trail leaders Real Madrid who have 10. The Danes are unbeaten at home in the Champions League group stage during their three campaigns.
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(Writing by Philip O'Connor; Editing by Ken Ferris)
