Brazilian Gustavo curled in an 18-metre shot in the 81st minute for the visitors and Mandzukic tapped in his sixth goal of the season two minutes later to lift Bayern to 18 points.
Borussia Dortmund bounced back with an emphatic 5-0 demolition of Borussia Monchengladbach and two goals from Marco Reus for their first win in three games to move up to third on 11 points.
"We were too slow in the first half and my players tested my patience a bit," Bayern coach Jupp Heynckes, whose team take on BATE Borisov next week in the Champions League, told reporters.
"I had to find hard words for my half-time talk and in the end it was enough for a win against a tough Bremen side."
The Bavarians have now won all matches in all domestic and European competitions they are in.
Champions Dortmund were eager to send a message to their in-form title rivals they were far from out of the race after failing to win their last two games.
Germany forward Reus, who joined Dortmund this season for 17 million euros, opened his account against his former team with a fine solo effort and defender Neven Subotic drilled in a header as the hosts struck twice in a five-minute spell late in the first half.
Reus added another, surprising keeper Marc-Andre ter Stegen with audacious shot from a tight angle in the 70th before late goals from Ilkay Guendogan and Jakub Blaszczykowski completed the rout.
The champions, playing Manchester City in the Champions League next week, are now up to third on 11 points.
Bremen did well to stifle almost all the Bayern attacks in the first half before Franck Ribery rattled the post with a shot from 14 metres after shaking off Clemens Fritz twice early in the second.
The hosts had a chance of their own with Nils Petersen, on loan from Bayern, but the striker failed to beat keeper Manuel Neuer with a volley.
Gustavo, who minutes earlier had seen his low drive saved by keeper Sebastian Mielitz, finally broke the deadlock, when unmarked at the edge of the box he curled in an unstoppable shot.
Before Bremen had any time to recover, the Bavarians struck again as France's Franck Ribery combined well with Xherdan Shaqiri on the left and the Swiss floated in a perfect cross for Croat Mandzukic.
Hoffenheim, stunned by a car accident which left midfielder Boris Vukcevic in an artificial coma after he sustained life-threatening injuries on Friday, shared the spoils in their goalless draw at home to Augsburg.
Fans held banners with the 22-year-old's name while Hoffenheim were given the option to postpone the match but after consulting with Vukcevic's parents they decided to go ahead.
"That was no normal game," Hoffenheim coach Markus Babbel told reporters. "The result is of no importance. I wanted to see a team that will fight for Boris and that is what I saw."
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