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.
Rooney put United ahead from the penalty spot soon after and, with Braga in disarray, Hernandez made it 3-1 in injury time after a goal-mouth muddle.
"Those last 10 minutes went really badly for us. What a shame," Braga midfielder Ruben Micael told Portuguese television.
Group H leaders United have 12 points from four games. Cluj and Galatasaray are on four points, with Braga on three.
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."
United coach Sir Alex Ferguson made several changes to the side who beat Arsenal 2-1 in the Premier League over the weekend, bringing in Nani, Anderson and Giggs to the midfield and picking Danny Welbeck to spearhead the attack with Hernandez, leaving Van Persie on the bench to start with.
Centre-back Smalling played for the first time since injuring his metatarsal in the pre-season.
The changes did not pay off and United were poor in the first half, coming to life only after Van Persie was sent on with 25 minutes on the clock.
Braga suffered a near-repeat of their fortunes at Old Trafford two weeks ago, when they gave away a 2-0 lead and lost 3-2.
"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.
