Zinnbauer – the club's former under-23 coach – replaced Mirko Slomka in the top job at Hamburg last month.
And while Zinnbauer's arrival coincided with a goalless draw against two-time defending Bundesliga champions Bayern Munich, a loss at the second-placed Borussia Monchengladbach was followed by a heartbreaking 2-1 defeat against Eintracht Frankfurt last week.
Hamburg – who beat second-tier Greuther Furth on away goals in last term's relegation play-off – dominated for much of their home clash against Frankfurt, only to lose to Lucas Piazon's stunning stoppage-time free-kick.
And a trip to Dortmund, who have finished in the top two for the last four years, means Zinnbauer's men will have to be at their best to claim their first win of the season.
Hamburg director Dietmar Beiersdorfer is convinced, that in Zinnbauer, that the club have the right man for the job.
"Joe is doing a very good job and is very emotionally involved," Biersdorfer told Hamburg's official website.
"You can see that something is stirring. He sends a team out after talking a lot of sense to them.
"The situation is difficult...we are in bottom place. Now everyone at the club and inside that dressing room must invest even more effort."
Hamburg will take heart from Dortmund's patchy recent form, which has seen them fail to win any of their last three league matches.
A 2-1 defeat to arch-rivals Schalke last week saw them drop to 12th – a position that is unlikely to last for long.
Hannover's bright start to the season has faded somewhat, with two losses in their last three, and they head to Munich to face a Bayern side who have hammered them in recent times.
Bayern have won the last six clashes between the two sides, scoring 23 goals in the process, while they have netted four or more against Hannover on six occasions since 2009.
Leaders Bayern are two points ahead of Monchengladbach, who host the fourth-placed Mainz on Sunday in a meeting between two of the four remaining unbeaten sides.
At the other end of the table, just one point ahead of Hamburg are Werder Bremen, who take on fellow strugglers Freiburg.
Stuttgart won for the first time last week and battle Hertha Berlin on Friday, while Cologne will aim to end a winless run of four against Eintracht Frankfurt.
Elsewhere, Bayer Leverkusen meet Paderborn, Hoffenheim face Schalke and Wolfsburg take on Augsburg.