Outstanding Dutch winger Robben scored with a spectacular free kick three minutes after Hugo Almeida had levelled for Bremen.
Bayern, who extended their winning streak to eight matches in all competitions, have 39 points from 19 games.
One point behind are undefeated Bayer Leverkusen, who can reclaim the top spot on Sunday against Hoffenheim, and Schalke 04, who squandered a 2-0 lead to draw 2-2 at VfL Bochum.
Schalke wasted their chance to go top when Vahid Hashemian (82 minutes) and Stanislav Sestak (stoppage time) cancelled out first-half strikes by Vicente Sanchez and Kevin Kuranyi.
Borussia Dortmund leapfrogged Hamburg SV into fourth place on 36 points, beating the northern German club 1-0 thanks to a goal from Paraguay international Nelson Valdez for their sixth straight win.
"Today I am a satisfied coach," Bayern's Louis van Gaal told reporters. "I liked that we were very aggressive and we used Bremen's game to our advantage.
"Unfortunately though we did not make the best of the many chances we had."
Bayern dominated throughout and hit the post after only three minutes through Robben, who went on to terrorise the Bremen defence, before Ivica Olic was denied by home keeper Tim Wiese.
The hosts, badly missing virus-hit striker Claudio Pizarro, took an unexpected lead on 10 minutes when Marko Marin sent Aaron Hunt into the box with a perfect pass and the striker fired an unstoppable left-foot shot into the top corner.
Bayern levelled 15 minutes later when Mario Gomez let a Bastian Schweinsteiger pass go through his legs for Thomas Mueller to score from 12 metres.
Olic made amends for his earlier miss when he tapped in a Philipp Lahm cross in the 35th and rampant Bayern should have added a third just before the break when Robben missed an open goal.
The visitors, who brought on playmaker Franck Ribery as a substitute after being out injured for three months, squandered a host of chances and were punished when Portuguese Almeida equalised for Bremen in the 75th.
Robben, though, had the last laugh when he whipped a fantastic free kick into the top corner.
"We gave Bayern so many chances," said Bremen coach Thomas Schaaf after his team's fourth straight defeat. "It was a great match for Bayern fans."
