Goalless draws with Hamburg and Cologne, followed by Wednesday's UEFA Champions League exit at the hands of Juventus, had hampered Dortmund's 2015 revival.
However, Aubameyang's double and a Shinji Kagawa tap-in punished Bittencourt's red card, with the Gabonese striker shaking off any early Dortmund rustiness when he latched onto Marco Reus' pass 19 minutes in to open the scoring.
It was Dortmund's first league goal since February's derby win over Schalke but Hannover were level just after the half-hour through Lars Stindl.
Stindl made the most of strong running from Bittencourt before the former Dortmund man was dismissed early in the second half for two mistimed tackles in quick succession.
Hannover were made to pay when Kagawa tapped home following good work from Jakub Blaszczykowski, with Aubameyang heading home the Japan playmaker's cross to secure the points – Stindl's late stunner doing little to ease pressure on under-fire Hannover coach Tayfun Korkut.
The hosts almost capitalised on a breakdown in communication between Neven Subotic and Roman Weidenfeller eight minutes in as Dortmund started slowly following Wednesday’s European exploits.
However, the game's first moment of quality brought about the opening goal as Blaszczykowski - making his first Bundesliga start since January 2014 - found Reus infield with the help of Sebastian Kehl's dummy.
Reus' clever pass played in Aubameyang, who calmly tucked away his first 11th Bundesliga goal of the season.
Having ended their three-game goalless spell, Dortmund initially looked to grow in confidence but were pegged back when Hannover broke after Blaszczykowski was dispossessed, and Bittencourt cut back for Stindl, who slotted the equaliser past Weidenfeller.
Hannover put Jurgen Klopp's men on the back foot late in the half and Bittencourt should have put them ahead when through on goal, but instead blazed wide.
Mats Hummels was required to clear Joselu's header off the line shortly after the interval before Bittencourt received his marching orders to turn the tide.
Two late challenges on Blaszczykowski and Kehl in the space of six second-half minutes gave Felix Brych no choice but to brandish a red with Dortmund quickly making the numerical advantage count.
Just two minutes later, Kagawa restored the visitors' advantage following good work from Blaszczykowski, before the Japan international turned provider for Aubameyang's diving header on the hour.
After Reus hit the post, Stindl scored his second - a dipping long-range strike nine minutes from time – and while it threatened a comeback, Klopp's side held firm to see out the win and move within five points of the Bundesliga's European spots.
