Dortmund bounced back from their Champions League exit to Juventus this week to move up to 33 points, five behind Augsburg, who are sixth and occupy the last place for Europa League qualification.
Dortmund went ahead with their first goal in four matches when Aubameyang netted in the 19th minute to score his 11th goal of the season.
They dropped their guard a little later, allowing Lars Stindl to level from close range.
Hanover were down to 10 men early in the second half when Leonardo Bittencourt, a former Dortmund player, got his marching orders following a second booking.
Dortmund made the extra man count instantly, with Shinji Kagawa tapping in after good work from Jakub Blaszczykowski and a Marco Reus assist.
The Japanese then chipped a ball into the box and Gabon international Aubameyang connected with a diving header for his second goal before Stindl also got a second to set up a nervous finale.
Werder Bremen suffered a setback in their European aspirations, conceding an 88th-minute equaliser for a 1-1 draw at Cologne that kept them a point ahead of Dortmund.
Davie Selke had put the visitors ahead before Matthias Lehmann's spot kick two minutes from time levelled the score.
VfB Stuttgart reignited their relegation battle, coming from a goal down to stun Eintracht Frankfurt 3-1 for their first victory in 10 games. Despite the result, they remain bottom of the standings but are within two points from safety.
Stuttgart looked to be dropping deeper into trouble after Haris Seferovic's strike in the 51st minute but two goals in three minutes from Daniel Ginczek turned the game around.
Alexandru Maxim made sure of the three points for the 2007 Bundesliga champions late in the game giving coach Huub Stevens some respite.
Bayer Leverkusen, in fourth place, take on Schalke 04, three points behind in fifth, later on Saturday.
Leaders Bayern Munich, 11 points clear at the top, host Borussia Moenchengladbach on Sunday, while second-placed VfL Wolfsburg travel to Mainz 05.
(Reporting by Karolos Grohmann, editing by Pritha Sarkar)
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