Dortmund were defeated in a hectic contest after Dan-Axel Zagadou was shown a straight red card for a trip on Jonathas, meaning reigning champions Bayern Munich were able to overthrow Peter Bosz's men at the summit by defeating RB Leipzig 2-0 later in the day.
Jonathas broke the deadlock from the penalty spot and then turned provider for Ihlas Bebou to restore the hosts' lead after Zagadou hit back with his first Bundesliga goal.
Andriy Yarmolenko struck a brilliant second equaliser, but Felix Kraus converted the free-kick that was awarded after Zagadou's dismissal and Bebou wrapped up the points with four minutes remaining.
After taking 19 points from their first seven games of the league season, Dortmund have now lost two and drawn one of their last three fixtures to open the door for a rejuvenated Bayern.
Christian Pulisic and Pierre-Emerick Aubameyang both recovered from injury concerns to start, but BVB were dealt a major blow in the 20th minute as the hosts opened the scoring.
Roman Burki upended Klaus as he stretched for the ball in the area and - after referee Patrick Ittrich used VAR - Jonathas tucked away the penalty.
Dortmund's response was strong, however, and the equaliser soon followed from the unlikely source of Zagadou, who volleyed beyond Philipp Tschauner as Hannover scrambled to clear a right-wing corner.
Bosz's men tore forward in search of a second and Yarmolenko should have supplied it, somehow blazing high and wide after Pulisic - a menace on the left - raced away from the bumbling Salif Sane.
A thrilling first half continued in its unpredictable vein, though, and Bebou restored Hannover's lead with a cool finish at the end of a lightning counter-attack.
There was again a Dortmund riposte when Sane met an outswinging cross with a tame header and Yarmolenko positioned himself beneath the dropping ball to send an effort arrowing into the bottom-right corner.
But the visitors' momentum was catastrophically disrupted once more as Zagadou tripped Jonathas and was duly dismissed, before Kraus exquisitely curled the resultant set-piece home.
Only a fine Tschauner stop kept Pulisic from netting another equaliser and Hannover could have widened their advantage as Bebou led a break and slipped in Kraus to shoot straight at Burki.
Dortmund's chief tormentor Bebou added to his tally when he streaked away again and lashed a finish inside the left-hand post.
Although Burki denied Hannover a fifth by keeping out Martin Harnik's late overhead kick, the hosts had already done enough to clinch an impressive win.
Key Opta stats:
- Dortmund have conceded at least two goals in three consecutive Bundesliga games for the first time in more than three years (September 2014 under Jurgen Klopp).
- Since October 26, 2013, all 12 Bundesliga penalties against Dortmund have found the net. Roman Burki has not saved any of the 11 spot-kicks against him, nine of which have come as a BVB player.
- In his ninth Bundesliga appearance, Dan-Axel Zagadou scored his first goal and later received his first card.