Bremen lost Milos Veljkovic to a second yellow card near the hour mark at the Allianz Arena.
"I wish that we would have used the multiple chances we had to get 2-0 or 3-0, but the 1-0 is enough," said Bayern coach Niko Kovac. "The win is fully deserved."
Bayern leads with 70 points and has four games left. Second-place Borussia Dortmund, which plays at Freiburg on Sunday, has 66 points.
Coming into the game, Bremen had not lost in any competition in 2019, and had scored in every one of its Bundesliga matches this season.
Also Saturday, third-place Leipzig won 2-1 at fifth-place Borussia Moenchengladbach. Sixth-place Hoffenheim defeated Schalke 5-2.
In other matches, Augsburg thrashed relegation-threatened Stuttgart 6-0 and Bayer Leverkusen beat 17th-place Nuremberg 2-0.
Mainz beat Fortuna Duesseldorf 3-1 in a mid-table clash.