Szabolcs Huszti kept his cool to score an 89th-minute penalty to earn the Reds all three points and end Augsburg's club-record three-game winning streak.
Paul Verhaegh also scored from the spot to give Augsburg the lead in the 51st minute before Artur Sobiech levelled matters on the hour mark.
The game took a while to warm up inside the Niedersachsenstadion with Hannover unable to express their usual dominant game on home soil.
Augsburg, buoyed by three straight wins, showed no fear, even if only four sides had managed to win in the Lower Saxony capital last season.
Towards the end of the first half, Leon Andreasen tested Augsburg goalkeeper Alex Manninger with a 25-yard free-kick, but the Austrian rose to the occasion to keep the ball out.
Augsburg's best chance was a volley from Sascha Molders which blazed over.
His next shot on goal struck the arm of Salif Sane, and the referee pointed to the spot, giving Verhaegh the chance to score from the spot, and he took it.
Hannover saw similar claims for a penalty waved away when Didier Ya Konan's effort struck Ragnar Klavan, but the referee this time did not award a spot-kick.
Sobiech then took advantage of a mix-up between Manninger and Klavan, who clashed in mid-air, leaving the ball for the Polish forward to tuck in.
Hannover searched for a winner and they found it late on when Ya Konan's header hit the arm of Matthias Ostrzolek, and this time a penalty was awarded.
Huszti stepped up and sent the ball into the right-hand corner to earn Hannover their fourth home win of the season.