Andre Schurrle scored after five minutes at Weserstadion to put Dortmund on course for their first Bundesliga victory in four attempts, and Bremen goalkeeper Jaroslav Drobny was sent off after 38 minutes when he brought down Marco Reus.
Despite taking a lead and a numerical advantage into the second half, Dortmund were pegged back by Fin Bartels' equaliser for Bremen, before Lukasz Piszczek scored the winning goal in the 71st minute.
The win moved Dortmund into the Bundesliga top four, but Tuchel's men still trail leaders Bayern Munich by 12 points.
The Dortmund coach told Sky: "We started very well, but we missed chances in the first 15 minutes to score the second goal.
"Then we recklessly allowed Bremen back into the game. The match was then completely open.
"We missed a number of chances and we were sluggish in the rest of the game, which has really annoyed me."
The victory came in the absence of Dortmund's top goalscorer Pierre-Emerick Aubameyang and was only the club's second in their last seven league away games, prompting Tuchel to admit: "The joy of the victory prevailed.
"Perhaps we need more success stories of this kind in order to bring back the easiness again."