With many tipping David Moyes' side to be on the end of a hiding in their quarter-final first leg, United ensured they go to the Allianz Arena next week with a fighting chance even though Nemanja Vidic's second-half header was cancelled out by Bastian Schweinsteiger - who was then sent off late in the game - on 66 minutes.
United would happily have taken a draw after spending much of the first half chasing shadows as Bayern dominated possession with Toni Kroos orchestrating matters in midfield.
The Germans enjoyed almost 80 per cent of the ball in the opening 45 minutes and came closest to scoring when Arjen Robben was denied by a superb one-handed save by David de Gea.
But it was possession-starved United who had the best chance when Danny Welbeck - who had a goal ruled out for a raised foot after only three minutes - burst clear of Jerome Boateng but instead of firing the ball beyond Manuel Neuer he looked to chip the giant German keeper who stuck out his right hand to save.
Moyes brought on Shinji Kagawa at half-time for the injured Ryan Giggs as United looked to build on the signs of encouragement late in the opening period.
And they gained reward on 58 minutes when Vidic found space in the Bayern box to head in Wayne Rooney's corner.
The lead lasted just eight minutes before Schweinsteiger punished the home side for failing to pick up his late run into the box, pouncing on substitute Mario Mandzukic's knock down to fire into the roof of the net.
Bayern ended the game with 10 men when goalscorer Schweinsteiger chopped down Rooney as the England striker looked to break away to earn a second yellow card which rules him out of the second leg along with Javi Martinez.
United will go to Germany confident they can create an upset after causing Pep Guardiola's side problems at Old Trafford - in spite of surrendering the lion's share of possession to their opponents for long periods of the game.
Welbeck promised some early joy when he rammed home a fierce strike from the edge of the box, but the whistle had already gone for dangerous play for a high foot.
That was but a brief interlude, however, in the wave after wave of Bayern attacks, orchestrated by Robben and Franck Ribery.
In contrast, United's passing was pedestrian, and Old Trafford's growing frustration with Marouane Fellaini came close to spilling over as the tall Belgian struggled and even found himself out-jumped for headers that he seemed clear favourite to win.
For all Bayern's superiority, there was always the suspicion that Boateng was vulnerable to Welbeck's pace.
The England striker had already hit another firm shot straight at Neuer before missing a great chance when one-on-one with the German keeper.
United looked threatening after the break with Welbeck and Rooney having greater support in attack and testing the Bayern defence before taking the lead through Vidic's guided header.
Guardiola responded by sending on Mandzukic for Thomas Muller, and the change paid dividends in the 66th minute as the Croatian striker headed back Rafinha's cross for Schweinsteiger to finish emphatically into the top corner.
Old Trafford, which had been re-energised by Vidic's goal, was suddenly deflated and it was Bayern who looked like finishing the stronger before Schweinsteiger saw red just moments after Antonio Valencia had escaped a similar punishment for a robust challenge on Boateng.