Braaten back-heeled the ball in to give FCK a fifth-minute lead and they withstood a second-half charge from Gala to claim the win which sees them climb off the bottom of Group B and level on four points with their opponents.
Real Madrid top the group with 10 points with Juventus now bottom with three.
The Danish champions came into the match at the Parken stadium with just one point to their credit and their hopes of a top-two finish seemingly fading, but they enjoyed the ideal start with Braaten's inventive close-range finish.
Rurik Gislason fired over a low cross from the right and Braaten ghosted in between the Gala backline and goalkeeper Eray Iscan before showing great audacity and no little skill to flick the ball inside the back post with his heel.
Stale Solbakken's troops then came within inches of making it 2-0 two minutes later when Nicolai Jorgensen saw his header from Pierre Bengtsson's free-kick hit the post.
It was looking set to be a long night for Gala at that stage, but they weathered the early storm and were almost on level terms in the 21st minute when Burak Yilmaz's header brought a brilliant save out of Johan Wiland.
The visitors came even closer 14 minutes later when they too hit the woodwork, Felipe Melo's header striking the upright from an Albert Riera corner.
FCK's attacking threat had not disappeared completely though and three minutes before the break they had a great chance to make it 2-0 when Bengtsson sent Braaten clean through but he was denied a second goal by a good save from Iscan.
There was still time after that for Roberto Mancini's men to create another chance before half-time and Didier Drogba may feel he should have done better with his close-range effort.
Drogba headed another opening wide at the start of the second half and the Ivorian, who led Chelsea to Champions League glory in 2012, then forced two more saves out of Wiland midway through the period as the Turkish side turned up the heat on their hosts.
Yilmaz was next to test the Wiland with a headed effort eight minutes from time, but the visitors would not get any closer in the closing stages as FCK held on for the win.