Cristiano Ronaldo was held scoreless for the first time in the competition this season.
The Portuguese star had an effort ruled out in the second half for using his arm to control the ball.
Bayern, chasing a treble in Jupp Heynckes' last season in charge, was hoping to make up for last year's quarter-final exit to Madrid, and for the Spanish side's 5-0 aggregate win at the same stage in 2014.
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Rafinha had the first shot on goal in the 19th minute after a cautious start from the visitors, who started with both Gareth Bale and Karim Benzema on the bench.
Madrid started to settle midway through the half before Kimmich struck.
James Rodriguez played the right-back through as Bayern created a three-on-two. Perhaps expecting a cross, Navas was surprised when Kimmich let fly to score inside the far corner.
Buoyed by the goal, Bayern's players suddenly found spaces but Ribery wasted a good opportunity to double the lead.
Instead, Marcelo levelled the score shortly before the break.
Bayern missed more chances before the interval, when Asensio came on for Isco to change the course of the game.
After Bayern turned the ball over in midfield, Vazquez and Asensio played a one- two before the substitute coolly dispatched the ball past Sven Ulreich.
Reuters
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