The Red Devils captain broke the deadlock in the 58th minute at Old Trafford after the Serbia international powered a header past Bayern goalkeeper Manuel Neuer.
However, United’s lead was short-lived, with Germany international Bastian Schweinsteiger scoring from close range to net a vital away goal in Manchester.
The 29-year-old midfielder was sent off after picking up a second yellow card for a reckless challenge on Rooney in the closing moments of the tie, with Javi Martinez also banned for the second leg.
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