Nemanja Vidic broke the deadlock in the 58th minute, heading home Wayne Rooney's corner-kick after being left unmarked by the Bayern defence.
But the Bundesliga champions restored parity and got what could prove to be an important away-goal when Bastian Schweinsteiger finished off Mario Mandzukic's knock-down on the bounce, sending the ball into the roof of the net with a swing of his left boot.
In the first half, Danny Welbeck was guilty of missing a glorious one-on-one opportunity in the 40th minute but Manuel Neuer stood tall in the Bayern goal to block his attempted chip.
Arjen Robben also had a chance in the 78th minute to grab another goal for the away side but dragged his shot wide with his weaker right foot when well placed to score.
The game ended on a sour note for Pep Guardiola's side when goalscorer Schweinsteiger was shown his marching orders for a second yellow card in the 90th minute, meaning he will miss the return leg in Germany.
