Burak Yilmaz cancelled out Mossoro's opener before Aydin Yilmaz tapped home with just 15 minutes remaining to send the visitors into the last 16
The Turkish champions appeared to be heading out when Mossoro gave their Portuguese opponents a fully deserved lead shortly before the half-hour mark.
The situation worsened for Gala when qualification rivals Cluj went a goal up on table-toppers Manchester United, but Burak Yilmaz and Aydin Yilmaz scored second-half goals to see Fatih Terim's men through in dramatic fashion.
Braga went into the game with nothing to play for at the Estadio AXA but they dominated right from the off, with Mossoro firing over after being picked out in the area by Alan before Ismaily called Fernando Muslera into action for the first time with a low drive.
The hosts should have taken the lead 15 minutes in, after Mossoro had slipped Eder in on goal, but the forward took far too long to get his shot away, allowing Felipe Melo to deflect the ball over the bar.
Just moments later, Gala were indebted to Dany Nounkeu for an equally important block, with the ball having dropped invitingly for Paulo Vinicius.
Braga continued to press as the half wore on and Ismaily, Eder and Alan all tried their luck before Mossoro showed his teams-mates how it was done 32 minutes in.
Braga attacked down the right-hand side and after Melo had completely missed his kick in trying to clear on the edge of area, Mossoro took a sublime touch to take him past his fellow Brazilian before curling the ball into the bottom corner.
The goal finally raised the visitors from their slumber and Selcuk Inan drew a fine finger-tip save from Quim with a 30-yard free kick from a central position but Braga still finished the half the stronger, giving Terim much to think about during the break.
The former Turkey boss decided he had seen enough from Johan Elmander and Hamit Altintop, replacing them with Aydin and Nordin Amrabat, respectively.
The visitors suddenly looked far more potent up front and they drew level just before the hour mark when Burak rose to nod in Amrabat’s wonderful inswinging cross from the left wing.
Ismaily very nearly put the hosts back in front with a venomous effort which flew just wide, but the momentum was now with the visitors and they nicked the victory which saw them progress to the last 16 when Aydin turned in the rebound after Quim was unable to keep out a shot from Quim.