Claudio Pizarro and Mario Mandzukic struck in the late stages of the game as the Bavarian giants recover from their mini-slump
Since being crowned league champions in record time last month, Pep Guardiola's men have drawn once and lost twice in their last three games after previously going 53 unbeaten in the Bundesliga.
Rock-bottom Braunschweig provided stern opposition in the first half, but Bayern eventually found the breakthrough in the final 15 minutes thanks to goals from Claudio Pizarro and Mario Mandzukic at Eintracht-Stadion.
Jan Hochscheidt went close to putting the home side ahead with a 17th-minute header before Pierre-Emile Hojbjerg smashed over for the Bavarian giants.
Braunschweig troubled Bayern throughout the first half, but the visitors looked far superior after the break and Pizarro finally gave them the lead in the 75th minute before substitute Mandzukic made sure of the win 11 minutes later.
Bayern winger Arjen Robben challenged his team-mates in the build-up to end their recent poor form and boost confidence ahead of their upcoming Champions League semi-final clash against Real Madrid, and he nearly got them off to a great start.
With 13 minutes played, Franck Ribery latched on to a pass threaded in behind the Braunschweig defence before squaring to the Netherlands international, but Matthias Henn charged back to poke away with an open goal at Robben's mercy.
Braunschweig almost capitalised on that let-off four minutes later, but Hochscheidt headed Omar Elabdellaoui's right-wing cross just wide of the left-hand post.
After a somewhat laboured first 20 minutes, Bayern started to take control of proceedings and should have taken the lead just before the half-hour mark.
Guardiola's men did well to keep possession inside the Braunschweig area and Robben eventually cut a pass back to the edge of the box, only for Hojbjerg – making just his second Bundesliga start – to blaze a wild effort over.
Chances were at a premium at both ends at the start of the second half as Braunschweig crowded their defensive third.
Bayern did not muster a meaningful shot at goal until the 61st minute, when Daniel Davari in the home goal comfortably dealt with Hojbjerg's long-range attempt.
With the champions' creativity lacking in the centre, Philipp Lahm attempted to seize the initiative with 66 minutes played and his excellent cross was met by Jerome Boateng, who could only head wide.
Bayern finally got themselves ahead when Pizarro pounced on Mario Gotze's clever pass into the area and the Peruvian emphatically blasted past Davari.
Mandzukic then added a second when he collected a long ball to coolly slot under the goalkeeper four minutes from time.
Defeat leaves Braunschweig bottom on 25 points with three games left as relegation looms, while Bayern will be relishing Wednesday's semi-final first-leg trip to Madrid.