The away side showed few signs of this season's struggles during the first half, hitting the post early on and then scoring in the 25th minute through Gondorf, as the performance belied their lowly league position of 16th.
Robert Lewandowski also had a day to remember as he scored two to become Bayern's most prolific non-German scorer in the Bundesliga with 94 goals.
Chile global Vidal has been a regular first-team fixture since arriving at Allianz-Arena in 2015 from Juventus, making 73 Bundesliga appearances and scoring 13 goals.
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Inevitably Bayern fought back and equalised four minutes before half-time, with Muller netting his third league goal of the season in style.
Bremen's fearless display meant Bayern only took the lead for the first time with an hour gone. But the visitors were denied what would have been a precious point when Lewandowski and Muller struck in the closing stages.
Bayern moved 16 points clear after 19 rounds of the league, though Schalke could reduce that to 14 with a win over Hannover in Sunday's late game.
Bremen went agonsingly close to breaking the deadlock in the seventh minute when Max Kruse rattled the woodwork with a header.
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Bremen grew in confidence, but the hosts drew level with a superb equaliser by Muller just before the break.
And the Chilean almost made a quick impact, forcing Jiri Pavlenka into an important save with a testing volley.
However, Lewandowski put Bayern back ahead by leaping to smash home Mueller's cross in the 77th minute as Bremen goalkeeper Jiri Pavlenka fumbled the header.
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