David Moyes' side were forced to sit back and soak up large swathes of Bayern pressure before unearthing the opening goal on 58 minutes when Nemanja Vidic rose to head Wayne Rooney's corner into the visiting net.
It was a fine goal from the United captain, but the lead lasted only eight minutes when Schweinsteiger pounced on a cushioned header by Mario Mandzukic to rattle a volley high into the net after a lovely cross by Rafinha.
Bayern had their away goal ahead of next week's second leg on April 9 at the Allianz Arena, but Pep Guardiola's European champions will be without the suspended Javi Martinez and Schweinsteiger.
Martinez was booked for a late challenge on Javier Hernandez while Schweinsteiger walked in the dying moments for a second booking after he clattered Wayne Rooney. He claimed Rooney had taken a dive.
United had enjoyed moments of good fortune moments earlier when Antonio Valencia somehow avoid a second yellow and a red for leaping into a poor tackle on Jerome Boateng. But there is very much life left in this tie with United suddenly contemplating the prospect of toppling Europe's leading side on their home patch.
3' - WELBECK GOAL DISALLOWED: Welbeck slams a volley into the Bayern net, but he is penalised for a high boot in winning the ball above Martinez. Probably right call, but better from United.
29' - DE GEA MAKES FINE SAVE: Robben with a stunning swerving effort from distance. Hit that well from 25 yards, but the keeper was at full stretch to turn it around the post.
39' - WELBECK BLOWS GLORIOUS CHANCE: Glorious chance for United. Welbeck through on Neuer. Should have scored. Tried to chip the goalkeeper, but straight at the visiting man.
58' - VIDIC NODS UNITED AHEAD: Rooney with the corner, and Vidic rises to head into the corner of the net. United leading 1-0 against the European champions.
66' - SCHWEINI EQUALISER: Schweinsteiger drills the ball into the net from the Mandzukic header down. And Bayern are level at 1-1. Fine finish.
71' - VALENCIA SOMEHOW AVOIDS RED: Valencia tries to win the ball after challenging Boateng. Had already been booked. Should have been sent off for that effort. Really poor tackle.
88' - SCHWEINSTEIGER IS OFF: Schweinsteiger is off for a second yellow card. Looked a decent tackle. But he is sent off. He is out of the second leg.
5 - No player has made more assists in this season's Champions League than Wayne Rooney (5).
15 - United have scored in each of their last 15 Champions League meetings with German clubs.
29 - No Manchester United player had more touches in the first half than Marouane Fellaini.
22% - Manchester United had just 22% possession in the first half but had more shots on target than Bayern Munich (2-1).
7 - Bayern Munich have won seven successive Champions League away games.
TALKING POINT: Can Manchester United reach the last four? Bayern will be favourites to progress in the second leg, but United remain well and truly in this tie. They need a goal in Germany, but stranger things have happened. They have a fighting chance. Which is good news for David Moyes after what was expected to happen here before this contest.
MAN OF THE MATCH Nemanja Vidic (Manchester United): Brilliant performance from the home captain to keep the constantly roving Bayern forwards quiet.
Manchester United: De Gea 7, Jones 7, Ferdinand 7, Vidic 8, Büttner 8, Valencia 6, Carrick 7, Fellaini 7, Giggs 6, Rooney 7, Welbeck 7..subs: Hernandez 6, Young 6, Kagawa 6
Bayern Munich: Neuer 7; Rafinha 7, Boateng 6, Martinez 6, Alaba 6; Schweinsteiger 6, Lahm 6 Kroos 6; Robben 7, Muller 6, Ribery 7..subs: Götze 6, Mandžukić 6
MATCH IN A TWEET
- Sports & Recreation
- Manchester United
- Bayern Munich
- Bastian Schweinsteiger
- Wayne Rooney
- Nemanja Vidic
- David Moyes