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Paul Pogba wasted a good opportunity for Juve at the end of the opening 45 minutes when he shot tamely at goal with his weaker left foot, while Gianluigi Buffon saved David Alaba's fierce shot soon after.
Bayern killed off the tie with an hour played as Mario Mandzukic headed in the rebound after Buffon had kept out Javi Martinez's initial effort.
Thomas Muller could have made it 2-0 on the night minutes later, but he blazed his shot over the crossbar as the visitors tightened their grip on the contest.
Bayern wrapped up the match on 90 minutes when substitute Claudio Pizarro collected the ball inside the area before firing low into the far corner.