VfL Wolfsburg won an astonishing match at VfB Stuttgart, striking in the 90th minute after the hosts had a penalty saved and missed an open goal two minutes earlier.
The two other promoted sides won as two goals from Dani Schahin helped Fortuna Duesseldorf win 2-0 at Augsburg on their return to the top flight after a 15-year absence while Eintracht Frankfurt beat Bayer Leverkusen 2-1
A majestic free-kick by Juan Arango secured Borussia Moenchengladbach a 2-1 victory over Hoffenheim while troubled Hamburg SV slumped to a 1-0 home defeat against Nuremberg.
The most dramatic match was at Stuttgart, where the home team's Vedad Ibisevic had an 88th minute penalty saved by Diego Benaglio then incredibly fired the rebound wide from three metres with the goal at his mercy.
Ibisevic was still shaking his head in disbelief when Bas Dost headed in Vierinha's cross to give Wolfsburg the points.
Bayern, haunted by last season's disappointments when they finished as runners-up in the Champions League, Bundesliga and German Cup, struggled against feisty Fuerth until Thomas Mueller fired them in front two minutes before halftime.
Mandzukic, signed from VfL Wolfsburg after his impressive performances for Croatia at Euro 2012, celebrated his first Bundesliga game for Bayern with the second and Arjen Robben completed the win.
Fuerth took part in the last edition of the old German first division in 1962/63 but were not among the 16 clubs invited to take part in the inaugural Bundesliga the following season and had never returned to the top flight since.
Fielding six players who had never played a Bundesliga match before, the hosts frustrated Bayern for half an hour but offered little threat of their own in attack in front of a noisy, capacity crowd of 18,000 at their compact stadium.
Bayern, who began with Bastian Schweinsteiger on the bench after a summer of fitness problems, Franck Ribery sidelined with flu and Mario Gomez injured, went ahead when Dante's header was cleared off the line and Mueller scored from close range from the rebound.
In the second half, the lively 20-year-old Shaqiri instigated several dangerous attacks, one of which ended with Mandzukic heading the second.
"I was very optimistic, they have great motivation, you could see it in training and in the preparation matches," said Bayern coach Jupp Heynckes. "It wasn't easy to start with, as we expected, but we controlled the match after the first goal."
Schahin, who helped Fuerth to promotion last season, made a dramatic debut for Fortuna, who have been as low as the fourth tier since they were relegated in 1997.
He was brought on in the 60th minute for his first top-flight match, fired Fortuna ahead eight minutes later and struck again in the 79th minute, both from chances set up by Australian Robbie Kruse.
Moenchengladbach, beaten 3-1 at home by Dynamo Kiev in the first leg of the Champions League playoff, recovered to open their league campaign with three points against Hoffenheim thanks to a starring performance from Arango.
The Venezuelan captain provided the cross for Mike Hanke to head Gladbach in front in the 33rd minute and grabbed the winner with a trademark free kick after Roberto Firmino equalised for Hoffenheim.
Hanno Balitsch gave Nuremberg a 1-0 win at Hamburg SV where coach Thorsten Fink's future is already the subject of speculation after they struggled against relegation last season and were knocked out of the cup by third division Karlsruher SC last week.
Freiburg and Mainz 05 drew 1-1 while titleholders Borussia Dortmund beat Werder Bremen 2-1 on Friday.