The midfielder bagged a second-half away goal to hand the initiative to the German side, despite Nemanja Vidic's brilliant header
The hosts were under the cosh for almost the entire match but captain Nemanja 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. That run looked set to continue when the Serbian produced a header of the highest quality from Wayne Rooney's corner.
United was unable to build on the momentum as the recently-crowned German champions equalized through Schweinsteiger just eight minutes later.
However, Pep Guardiola will be without Schweinsteiger for the second leg after the midfield lynchpin picked up a second yellow card late for a foul on Rooney.
Danny Welbeck thought he had opened the scoring early on, only to be penalized for a high foot in the build-up.
After a fairly even opening 10 minutes, Bayern began to assert its 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 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 was made to pay for its 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.
The Red Devil joy was short-lived, 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 favorites to progress, although Moyes can take heart from the recent performances of Premier League rivals Manchester City and Arsenal at the Allianz Arena