The Norwegian scored a superb backheel after just six minutes to earn maximum points for the home side and blow Group B wide open, with Juventus now bottom of the standings
The Norwegian turned home a brilliant backheel after just six minutes to stun Roberto Mancini's men and seal the Danish champions' first victory of the Champions League this season and prolong their unbeaten run at home.
Both sides hit the woodwork in a frantic first half before a more cagey second period in which there was little goalmouth action, as Copenhagen maintained their record of never having lost at home in the group stages.
The win leaves Group B wide open, with Stale Solbakken's side now level with their opponents on four points, with Juventus a point behind them at the foot of the table with two matches remaining.
Solbakken made one change from the weekend as Igor Vetokele was replaced by Youssef Toutouh, despite scoring twice in the 4-0 win over Nordsjaelland on Saturday, while the visitors made four changes from their last outing, with Wesley Sneijder (thigh) and goalkeeper Fernando Muslera (toe) among those missing out.
The hosts made a dream start as Braaten put them ahead inside six minutes, getting across Emmanuel Eboue in the six-yard box to backheel Rurik Gislason's low cross into the net.
And that advantage was almost doubled two minutes later as Nicolai Jorgensen, who also netted a double at the weekend, headed Pierre Bengtsson's right-wing free-kick against the upright after Albert Riera had brought down Lars Jacobsen.
After a breathless start, Galatasaray eventually got a foothold in the game, Bruma having his close-range effort disallowed for offside after Johan Wiland had brilliantly denied Burak Yilmaz's downward header in the 21st minute.
Felipe Melo then went close to an equaliser for Mancini's side 10 minutes before the break, thumping a header against the post from Riera's corner before nodding the rebound over the crossbar.
As the half drew to a close, both sides missed golden opportunities to score within a minute of each other.
First, Bengtsson released Braaten, who charged half the length of the field only to be denied by Eray Iscan, before Didier Drogba volleyed Aydin Yilmaz's delivery over the crossbar from eight yards out when totally unmarked.
After a quiet start to the second period, Drogba stung the palms of Wiland with a fierce drive in the 69th minute, before the goalkeeper made a more straightforward save from the Ivorian five minutes later.
The visitors continued to push hard for an equaliser in the final stages, but Copenhagen stood firm to claim their first win of the campaign and put themselves back into contention for qualification for the last 16.