Maxime Lestienne (51) headed Club Brugge in front in Germany, but Artur Sobiech (73) equalised before Jan Schlaudraff (80) won the game with an audacious chipped penalty.
The sides have been paired for the first time in UEFA competition.
Club Brugge’s record in 24 games against German sides is W8 D5 L11 (W8 D2 L2 at home).
Hannover lost 2-0 and drew 0-0 against R. Standard de Liège in their first games against Belgian opponents in this season’s UEFA Europa League group stage.
Club Brugge are without a win in their last two European home games – a defeat and a draw – since opening their 2011/12 campaign with three home successes without conceding.
The Belgian side are competing in the round of 32 for the third time since the advent of the UEFA Cup group stage in 2004/05; they lost to AS Roma in 2005/06 (1-2 home, 1-2 away) and Valencia CF in 2009/10 (1-0 home, 0-3 away).
A 2-0 loss at Standard in their most recent European away game was Hannover’s first defeat on the road in Europe this season – and their first in continental action since going down 3-1 at SV Werder Bremen in the 1992/93 UEFA Cup Winners’ Cup first round – their first UEFA competition game.
Hannover are competing in UEFA competition for only the second time in their history, and this is the first time they have played in a post-Christmas club competition fixture. Their previous campaign, in the 1992/93 UEFA Cup Winners’ Cup, ended in a 4-3 aggregate defeat against Werder.