Arjen Robben and Mario Mandzukic gave last season's treble winners a flying start, but an own goal from Dante gave a plucky Gladbach hope before half-time.
Thomas Muller missed a penalty before David Alaba assumed spot-kick duties to score Bayern's third and chalk up the first three points of the season.
The game appeared to be decided after just a quarter of an hour as Bayern raced into an ominous two-goal lead.
Robben netted the first goal of the Bundesliga season in the 12th minute when he latched onto Franck Ribery's lofted pass and took just one touch to divert the ball past the onrushing Marc-Andre ter Stegen.
Less than four more minutes were played before Mandzukic made it two, tucking a loose ball in after Ter Stegen only parried Robben's free-kick, which Ribery got the slightest of touches to.
All was going according to plan for Bayern but Gladbach did not appear to get disheartened and, with a counter-attack symptomatic of the kind which led them to a fourth-placed finish in 2012, they almost replied immediately with Raffael missing the target.
The same player shot straight at Manuel Neuer and then set up Max Kruse to test the Germany number one as the visitors sought a foothold in the game.
Bayern defender Dante scored eight goals for Gladbach before moving to Bavaria, and he took that tally to nine for the Foals when he lunged to turn Juan Arango's cross inside his own post five minutes before half-time.
Ter Stegen ensured Gladbach went in at the interval just a goal behind by tipping Toni Kroos' effort over the crossbar before making another fine one-handed save to deny Mandzukic from the resulting corner.
The Gladbach goalkeeper was beaten on the hour mark, but the post came to his rescue to deny Kroos before Neuer's Germany back-up earned his stripes by denying Thomas Muller from the penalty spot after Alvaro Dominguez handled the ball.
The ball did not go out of play from Ter Stegen's one-handed save and Dominguez again handled the ball as he tried to get in Robben's way, resulting in a second penalty in a minute.
This time Alaba stepped up and swept the ball low into the bottom left-hand corner to seal Bayern and Guardiola's first win of the season.