Strikes from Arjen Robben, Mario Mandzukic and David Alaba secured a win on home turf, despite Dante's own goal, that kicked off their campaign in winning fashion
The Bavarians began in style and quickfire goals from Arjen Robben and Mario Mandzukic ensured Pep Guardiola's reign retained the element of domination that Jupp Heynckes had fashioned throughout last season's treble success.
A Dante own-goal just before half-time ensured that the Allianz Arena tie remained in the balance and Lucien Favre's men came out fighting after the break and threatened to equalise on a number of occasions.
However, Alvaro Dominguez conceded two penalties in the space of two minutes - both handballs - and, though Thomas Muller saw the first brilliantly saved by Marc Andre ter Stegen, David Alaba smoothly converted the second attempt in the 69th minute to put to bed any comeback from the visitors.
In Guardiola's first Bundesliga game, he was forced to leave out two of the club's close-season acquisitions with Mario Gotze still recovering from injury and Thiago Alcantara suffering from illness.
Recent signings Christoph Kramer, Raffael and Max Kruse started for the visitors, with Lucien Favre having led his side to a draw and victory in their previous two visits to the Allianz Arena.
Gladbach were put on the back foot immediately as Bayern swarmed forward in the early stages, and the hosts were rewarded for their pressure in the 12th minute.
Franck Ribery lofted a well-executed pass over the top of the visitors' defence, with Robben racing onto the ball and prodding the ball past Marc-Andre ter Stegen in goal.
Bayern's dominance was reinforced just four minutes later as Mandzukic double his side's lead, pouncing on a rebound after Ter Stegen had kept Ribery out from Robben's free-kick from the right.
Favre's side still managed to force chances with Manuel Neuer forced into a smart reflexes stop by Raffael, but struggled to contain the constant wide threat of Ribery and Robben.
However, Gladbach were given a route back into the game, shortly before the break thanks to a Dante own-goal.
Juan Arango played the ball across the Bayern goal from the left flank with Dante poking past Neuer as the keeper came to claim the ball just behind him.
Ter Stegen was then forced into two wonderful saves to keep the score at 2-1, first tipping a Toni Kroos effort before denying Mandzukic's header from the resulting corner.
The second half began frantically, with Kruse having two good opportunities to equalise within 11 minutes.
Kroos then hit Ter Stegen's left-hand post from long range after 59 minutes before Bayern got the crucial breakthrough in a crazy three-minute period.
Alvaro Dominguez handled the ball in the box in the 66th minute but Muller saw his effort saved. However, bizarrely, Dominguez handled the ball again immediately after Ter Stegen's save and a second penalty was awarded.
Alaba took over duties and scored to restore Bayern's their two-goal lead, in the strangest of circumstances.
Despite the blow of conceding the third goal, Gladbach still threatened in the remainder of the game, the bright Kruse just being smothered by Dante as he raced towards goal.
However, they were unable to breach the hosts' defence and threaten a comeback, as Bayern took all three points against their opponents, just as they did in the final game of last season at Borussia Park.