Bayern leads Group C with 15 points, 5 points ahead of Italian Inter Milan, who also reached the knockout stage, after beating Viktoria Plzen 4-0 earlier, a result that knocked Barcelona out of the group stage for the second year in a row.
With the result not important for Barcelona, Bayern did not take long to open the scoring through Mane 10 minutes into the match, after receiving a wonderful long pass from Serge Gnabry.
Sane beat Spanish defender Hector Bellerin in the 50-meter race, and scored superbly.
Bayern doubled their lead in a similar way after 21 minutes, in light of the lack of vigilance and poor positioning of the Barcelona defense.
And Joshua Kimmich passed to Gnabry, who sent the ball from the edge of the penalty area to Chupo Moting, to score his sixth goal in the Champions League this season.
Bayern reduced their pressure after the break, but there was plenty of time for Pavard to score in the last attack of the match with a close shot from a corner kick.
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