In a match interrupted for 10 minutes by floodlight failure, the hosts went ahead early in the second half through an Alan penalty kick.
United struck back when Van Persie came off the bench and punished a blunder by Braga keeper Beto, who inexplicably rushed out of his goal and was caught in no man's land.
"Those last 10 minutes went really badly for us. What a shame," Braga midfielder Ruben Micael told Portuguese television.
Midfielder Ryan Giggs said United had made up for last year's failure to progress from the group stage. "I think we were all disappointed [last year]," he told Sky Sports. "It was not good enough and we knew we had to do something about it and we have. Long may it continue.
"We have really good strength in depth at the moment and it shows with Chris Smalling playing his first game in a long time tonight. It is good to have that strength when you have two games in a week and it is looking good for us."
"It's sad and we did not deserve it but we made mistakes and games are settled like that. Still, we played a really good match, even better than at Old Trafford," Braga coach Jose Peseiro said.
