Hannover substitute Mame Diouf scored five minutes from time to extend the hosts' unbeaten run at home to 22 straight matches and deny Dortmund, who had scored with Robert Lewandowski, their first back-to-back league wins this season.
The result means Dortmund, who also had three players injured in the game, drop to fourth on 12 points.
"Fouls were being awarded against my very clean team almost every second," Dortmund coach Jurgen Klopp told reporters. "Everyone needs ice packs - tendons, ligaments, everything."
"We should have finished off the game with our quick breaks. But I am not too interested in our gap with Bayern. I am more interested in the two games we still have to play against them this season."
They had keeper Roman Weidenfeller to thank for denying Hanover a fifth-minute lead after saving Artur Sobiech's long range effort.
Dortmund, who had to substitute defender Mats Hummels, Blaszczykowski and Sven Bender because of injuries, responded with a Pizszczek shot on the post before Weidenfeller's spectacular save on the hour from a reckless Piszczek clearance.
With Hannover piling on the pressure, Mame Diouf is unlikely to forget his missed chance when with 12 minutes left he found himself in front of an empty goal but miskicked from a metre out to send the ball into Weidenfeller's arms near the post.
Borussia Monchengladbach ended promoted Frankfurt's surprise unbeaten run in the Bundesliga for their first win since August.
Venezuelan Juan Arango's long-range thunderbolt and Dutchman Luuk de Jong's 25th-minute strike were enough to end Gladbach's five-game winless streak stretching back to
