Needing just a point to assure their qualification to the last 16, United fell behind to a penalty from Alan just after half-time but strikes from Robin van Persie, Wayne Rooney and Javier Hernandez ensured the Red Devils maintained their perfect record in this year's competition.
United lead the group on 12 points after four matches, while defeat for Braga leaves them bottom on three points, one behind both Galatasaray and Cluj following the Turkish champions 3-1 win in Romania.
Alex Ferguson elected to make six changes to the side that defeated Arsenal at the weekend, with Chris Smalling making his first start of the campaign after recovering from a metatarsal injury suffered during pre-season, while Robin van Persie was only named among the substitutes.
Jonny Evans was one of five holdovers from Saturday's victory over Gunners, but his misjudgement of Douglao's over-hit pass allowed Eder an early chance, the striker sneaking in behind the defence only to see his low strike flash across the face of goal.
More hesistant defending, this time from Smalling, then enabled Ruben Micael to run at United's backline and fire just wide from the edge of the box.
With the English giants very much on the back foot, Braga were unfortunate not to take the lead midway through the first half as Eder's diving header cannoned back off the foot of the post.
Alan, who scored twice against United in a 3-2 defeat at Old Trafford a fortnight ago, then forced a smart low save from David de Gea.
However, United's resistance was broken four minutes after half-time as Braga were awarded a penalty after Evans blocked off Custodio just inside the box.
Alan stepped up and duly converted for his fourth goal of the group phase.
But the game was interrupted only a few minutes later as a generator broke down, leaving the stadium shrouded in darkness and the players ordered back to the changing rooms.
With power restored roughly 10 minutes later, play resumed and only a sprawling save from De Gea kept out Nuno Andre Coelho's free-kick.
Braga, winners of their past four ties against English opposition on home soil without conceding a goal in that time, had been relatively untroubled for most of the evening but the introduction of Van Persie for Danny Welbeck proved to be a tactical masterstroke from Ferguson.
The Dutchman capitalised on a rash decision from Beto as he rushed off his line only to be left stranded as Van Persie clipped the ball over the Braga goalkeeper to draw United level with 10 minutes remaining.
The visitors then went ahead after Rooney slotted home from the spot after the England striker was felled by Coelho.
A late third from Hernandez completed United's turnaround, extending their unbeaten run in Portugal to five matches, leaving Braga to rue their late collapse.