
The three-time European champions needed both to beat Bayern in their group stage finale on Tuesday and hope for Copenhagen’s game against visiting Galatasaray to end in a draw.
Neither happened as Kingsley Coman slotted home from close range in the 70th minute for Bayern, who finished unbeaten in Group A to finish with 16 points.
United have just four points and their fourth-place finish means not even a Europa League playoff berth as consolation.
Copenhagen won 1-0 in Denmark to finish with eight points and reach the Champions League knockout round, while Galatasaray took the Europa League playoff spot in third with five points.
United won one only of their six group games, with their hopes of advancing hanging by a thread after they twice gave up a two-goal lead in a 3-3 draw at Galatasaray on Nov 29.
Bayern, who beat United 4-3 in their previous meeting in Munich, had already secured top spot in the group.