Kane sent goalkeeper Michael Zetterer the wrong way in the 56th minute and deep into stoppage time when he scored into the bottom left corner for a league leading 21 goals, after hitting the outside post and crossbar earlier in the game.
Both penalties were caused by Bremen's Anthony Jung, the first for handball and the second for tripping Jamal Musiala.
Substitute Leroy Sane scored Bayern's second in the 82nd at the far post.
After a difficult first half Bayern will be happy with the victory and a first clean sheet in six matches ahead of a Champions League play-off duel with Scotland's Celtic, with a Bundesliga topper at second-placed Bayer Leverkusen sandwiched in-between the two European matches.
"It's another nice game. You need moments to break the deadlock. As a striker you take responsibility," Kane told streaming portal DAZN.
"We have to keep that standard as high as possible. It's a big month for us with two big away games next week."
Bremen forward Marvin Duksch said: "We knew what kind of game we would face. We tried our best and defended well until the first goal."
He also maintained "we wouldn't have been given the penalty" over the handball incident.
Bayern dominated the first half of the Bundesliga-record 116th duel with Bremen, but found it hard to threaten the firm defence of the visitors who had won five of their previous 10 season games on the road, and 1-0 in Munich last season.
Kane came closest on the stroke of half-time when his header grazed the outside left post, after Kingsley Coman's earlier shot landed on the roof of the net.
Kane hit the crossbar seconds after the restart but was then his ruthless self from the spot.
He missed a second goal shortly after from a promising right angle as Bayern were now more dangerous against a harmless Bremen side coached by assistant Patrick Kohlmann because of Ole Werner's red card last week.
Sane scored off Konrad Laimer's low cross from the right, a minute after coming on for Coman; and Kane then chose the same corner as his first penalty on the last play of the match for the final score.
The other eight games are on the weekend when Leverkusen can't afford to drop points at Wolfsburg in the title race, and Niko Kovac makes his Borussia Dortmund coaching debut against VfB Stuttgart.
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