The Belgium international, on loan at BVB from Chelsea, was dropped to the bench for the Bundesliga home game after five games without a goal in all competitions.
Batshuayi came on with Dortmund a goal up thanks to a Marco Russ own goal and struck twice to seal victory in a frantic end to the encounter at Signal Iduna Park.
"I'm very relieved because I have not scored in recent games and have been able to help my team in the Champions League fight," Batshuayi told Goal.com after the game. "It was a very important victory against a team that had taken as many points as we had.
"For my goal to win the game I thought at first, 'I am offside', was briefly irritated, then I just shot, and it went in.
"I was not disappointed that I did not play from the beginning. The coach decides. I was tired after the last games, so I came into the game fresh."
The result saw Dortmund climb to third in the Bundesliga table.