After seven away victories in the Champions League, the title defenders came to Old Trafford as favourites.
They dominated possession but failed to create clear-cut chances and fell behind in the 58th minute when defender Nemanja Vidic headed in from a corner.
Bastian Schweinsteiger equalized in the 67th minute with a volley from a well-worked Bayern move.
But the midfielder was sent off just before the end for a second yellow card following a lunging tackle on Wayne Rooney.
However, his away goal means United will likely have to beat Bayern Munich away next Wednesday.
"We have a good starting position for the second leg," said Bayern captain Philipp Lahm. "The away goal is very important."
Goalkeeper Manuel Neuer added: "It's good that we came back after going behind. In Munich, we want to win, that is clear, but we know that we need to think about how we create clear goal scoring chances."
Bayern Munich chairman Karl-Heinz Rummenigge praised the German champions and said they were "very satisfied".
"I think 1-1 is a great result,” he said.