The Turkish side knew only a win would definitely be enough against their already-eliminated opponents, and their chances looked slim when Marcio Mossoro put the hosts in front just past the half-hour mark.
But a second-half equaliser from Burak Yilmaz and the winner from his namesake with 11 minutes to go sealed Galatasaray's progress at the expense of rivals CFR Cluj, despite the latter's excellent win at Old Trafford over Manchester United.
Braga had started like they had something to prove, with Mossoro firing over the bar after three minutes and Ismaily bringing a save out of Galatasaray goalkeeper Fernando Muslera two minutes later.
Braga came close again in the 14th minute when they forced a succession of corners, the last finding the head of Ismaily in the box who brought another save out of the busy visiting keeper.
Both Alan and Antonio Eder shot over the bar before Braga took a richly deserved lead when Mossoro got on the end of a clearance by Nuno Andre Coelho and fired a 20-yard effort past Muslera.
Selcuk Inan had Galatasaray's first real chance in the 39th minute with a 25-yard strike that was tipped over the bar by Braga keeper Silva Quim, but immediately at the other end Muslera was forced to save from Alan.
The Turkish side looked brighter at the start of the second half and equalised in the 58th minute when Burak Yilmaz got his head on a cross by Nordin Amrabat and nodded the ball past a helpless Silva.
Braga continued to threaten with Ismaily drilling a shot just wide, but Galatasaray grabbed the lead with 11 minutes left when Quim parried a shot by Felipe Melo and Aydin Yilmaz followed up from 12 yards.
The visitors still had to come through some nervy moments to book their passage into the knockout stages, but could have made it 3-1 in injury time when Burak Yilmaz drilled a shot against the post from 12 yards.