With the match level at 1-1 early in the second half, Hannover's former Dortmund midfielder Leonardo Bittencourt was shown a second yellow card, handing the visitors the initiative.
Shinji Kagawa and Pierre-Emerick Aubameyang then netted two quick goals to seal victory for Dortmund - who faced a tense final eight minutes after Lars Stindl pulled a goal back.
"[At half-time] I told my team that it would be cool if we could start to play football, but Hanover were still there on fire," said the head coach.
"But then came the red card, and then we made it great. The second and third goals were great, it would have been nice to get a fourth goal but we did not do it.
"We are now eight points clear [of the relegation zone] and are no longer in the mire."
But Aubameyang - who netted twice - has not given up hope of a top-six finish, and a place in Europe next season.
"There are only a few points now between us and sixth place and qualifying for the Europa League," he said. "We have to keep looking up the table and remain positive.
"Now, it's about taking as many points as possible."
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