The midfielder bagged a second-half away goal to hand the initiative to the German side, despite Nemanja Vidic's brilliant header
Nemanja Vidic opened the scoring with a brilliant second-half header but United's shock lead did not last long, with Schweinsteiger's crisp volley soon levelling proceedings.
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He will, therefore, miss the second leg on April 9, along with his defensive team-mate Javi Martinez, who picked up a booking of his own for a cynical foul on Javier Hernandez.
The hosts were under the cosh for almost the entire match but captain Vidic grabbed the opening goal 13 minutes into the second half as Europe's premier club competition continued to represent a bright spot in United's otherwise troublesome campaign.
Despite their fall from grace domestically, the Premier League champions had won all four of their European ties at Old Trafford this season prior to Tuesday and that run looked set to continue when the Serbian produced a header of the highest quality from Wayne Rooney's corner.
Vidic, departing for Inter at the end of the season, gave United the platform for victory but they were unable to build on it as the recently-crowned German champions equalised through Schweinsteiger just eight minutes later.
The away goal will make life that bit harder for United when they make the trip to Munich, with Pep Guardiola's side holding a slight advantage as the Spaniard looks to replicate predecessor Jupp Heynckes in securing a treble.
However, Guardiola will be without Schweinsteiger for the second leg after the midfield lynchpin picked up a second yellow card late on for a foul on Rooney.
Danny Welbeck thought he had opened the scoring early on, only to be penalised for a high foot in the build-up.
After a fairly even opening 10 minutes, Bayern began to assert their quality, dominating possession and creating a succession of half-chances.
However, the hosts stood firm and did well to deny their opponents any clear sights of goal.
Franck Ribery and Arjen Robben were involved in much of Bayern's best work but even they were unable to manufacture an opportunity until the Dutchman brought a stunning save from David de Gea with a curling left-footed effort from just outside the penalty area.
Welbeck then tested Manuel Neuer from range and went on to waste the best chance of the half after being put through one-on-one.
The England international attempted to beat Bayern's goalkeeper with a cheeky lofted effort, but Neuer was not fooled and palmed the ball away with his right hand.
The second half continued in much the same vein, with Bayern keeping possession and probing to little effect.
Bayern were made to pay for their inability to break down United when Vidic headed his side in front - angling a header past Neuer after Rooney had delivered a corner from the left.
Their joy was shortlived, however, as Schweinsteiger rifled home a knockdown from substitute Mario Mandzukic to bring the atmosphere at Old Trafford down a notch.
While Guardiola's men were unable to clinch a second away goal, they will remain heavy favourites to progress, although Moyes can take heart from the recent performances of Premier League rivals Manchester City and Arsenal at the Allianz Arena.
City won 3-2 in Munich in this season's group stage while Arsenal have picked up a draw and a win on their last two visits to Bavaria.