Forward Mandzukic volleyed in the winner after 13 minutes but despite having the lion's share of possession Bayern failed to convert a number of good chances.
Toni Kroos caught the Frankfurt defence off guard with a powerful header hitting the post as textbook pressing from the visitors quickly gave them control of the game.
Mandzukic did better in the 13th minute, drilling in a wonderful volley for their second win in two league games.
VfL Wolfsburg, with new signing Luiz Gustavo making his first start a day after joining from Bayern Munich, stunned Schalke 04 4-0 including three goals in 12 minutes.
Schalke's Klaas-Jan Huntelaar rattled the post with a fine low drive but Wolfsburg struck four times in the second half with Robin Knoche tapping in from close range and Vieirinha shaking off two defenders to fire past keeper Timo Hildebrand.
Defender Naldo added another in the 67th and Stefan Kutschke completed their first win of the season on the final whistle.
Hoffenheim, who needed a relegation playoff to stay up last season, crushed Hamburg with the visitors scoring four times in a dazzling second half and Frenchman Anthony Modeste grabbing a brace.
Champions League competitors Bayer Leverkusen were lucky to escape with a 1-0 win from Stuttgart with former Leverkusen player Daniel Schwaab netting an own goal.
Borussia Dortmund, last season's runners-up, host Eintracht Braunschweig on Sunday.
Image: Mario Mandzukic
Photograph: Alex Grimm/Getty Images