The best of those first-half efforts came after 31 minutes when winger Arjen Robben was allowed too much space and De Gea had to dive full length to beat away his shot.
Despite being overrun by Bayern’s attacking momentum, United had the best chance before the break when Rooney found fellow England striker Welbeck with a low through-ball after 40 minutes, which Bayern defender Jerome Boateng attempted to block, but fell over instead. That allowed Welbeck time and space to beat Neuer, but his weak chip gave the goalkeeper the chance to paw the ball safely away.
Ryan Giggs, making his 141st Champions League appearance, one short of Raul’s record, suffered a groin injury and had to be replaced by Shinji Kagawa at halftime.
The enterprising Japanese midfielder, who never lost to Bayern when he was at Dortmund, caused the visitors plenty of problems with some intelligent runs and passes.
From one of them United won a corner, which Rooney swept into the penalty area. The ball went slightly behind the incoming, but unmarked, Vidic, who turned his head to nod past Neuer for his first goal in Europe for more than five years.
United only led for nine minutes, though, before Schweinsteiger replied.
Dutchman Robben came close to a late goal for Bayern, but despite only drawing, the tie is balanced in their favor.