The club's former under-23 coach took over this week from Mirko Slomka, who paid the price for a poor start to the season that saw Hamburg rooted to the foot of the table with just one point.
However, Zinnbauer doubled that tally in his first game at the helm and will surely have been pleased with what he saw from his new charges, who proved more than a match for the champions.
Nicolai Muller came close for the hosts shortly after half-time, beating an onrushing Manuel Neuer to a high ball before guiding his shot just wide of the post, while Philipp Lahm spurned Bayern's best chance shortly afterwards.
The visitors, who recorded an impressive 1-0 UEFA Champions League over Manchester City in midweek, struggled to assert any form of dominance for much of the game, despite enjoying the vast majority of possession against a resolute Hamburg side.
Zinnbauer was quick to make his mark after being appointed as he stated that he would not be afraid to drop any players judged not to be pulling their weight, but he had little cause to complain on Saturday.
Having conceded more goals than any other Bundesliga side last season, the hosts were widely tipped to be steamrollered by Pep Guardiola's team, but did well to restrict the attacking talents of the visitors, who were rocked by an injury to Arjen Robben during the warm-up.
A much-changed Bayern came into the game having only tasted defeat once on the road in their last 37 attempts, but struggled to turn a lion's share of possession into any sort of attacking coherence.
Guardiola's men had to wait until the 20th minute for their first genuine opening, with Claudio Pizarro steering a volley wide from eight yards after a well-worked Rafinha cross.
Juan Bernat did manage to hit the target 13 minutes later, stinging the palms of Hamburg goalkeeper Jaroslav Drobny with an angled 30-yard drive, but that was as good as it got for Bayern during a frustrating opening 45 minutes.
The hosts nearly took an unlikely lead five minutes after the restart, as Muller beat both Jerome Boateng and Neuer to a high ball just outside the area, before diverting his effort agonisingly wide of the open goal.
Bayern predictably responded six minutes later, with Drobny on hand to deny Lahm after some good work from Xherdan Shaqiri inside the area.
Robert Lewandowski and Mario Gotze were thrown on midway through the second half as Guardiola sought to overload the Hamburg defence, and he nearly reaped the rewards 15 minutes from time.
A great touch form Thomas Muller inside the area allowed the forward to find enough space for a half-volley, only for Johan Djourou to deflect wide with a great last-ditch block.
Lewis Holtby could have won it for Hamburg five minutes before the end, but could only fire over after a bursting run following Bayern loanee Julian Green's lay-off.
Neuer survived a late scare when he appeared to handle the ball a long way outside of his area following a heavy touch, but he escaped with only a yellow as the spoils were shared.