Goals from Henrikh Mkhitaryan, Adrian Ramos and Pierre-Emerick Aubameyang completed the hosts' dramatic late comeback, lifting them to 54 points, eight points behind champions Bayern who moved to 62 with a 2-0 win at VfL Wolfsburg on Saturday.
"I am happy we kept calm and stayed patient," Dortmund coach Thomas Tuchel told reporters. "But we still managed to play a quick game and kept increasing the pressure.
"This was a hugely important win for us and we have to be satisfied with what the team delivered."
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Dortmund only fashioned their victory after Hoffenheim's Sebastian Rudy had turned from hero, following his 25th-minute goal, to villain with his sending-off 13 minutes after halftime.
Tuchel's men went behind when Rudy pounced on a weak save from keeper Roman Buerki to drill the ball in from close range.
Striker Kevin Volland almost doubled Hoffenheim's lead three minutes later but this time Buerki stood his ground to save a powerful shot.
Tuchel brought on playmaker Ilkay Guendogan after the break and he nearly scored within three minutes of his introduction, rattling the post after dribbling past three opponents.
Rudy then brought down the charging Aubameyang in the centre circle and was shown a straight red card.
It was a signal for Dortmund to turn up the heat but they had to wait until the 80th minute to make it count with Guendogan sending Armenian Mkhitaryan through to equalise.
Colombian Ramos, who had come on in the 74th minute, made it 2-1 five minutes from time, heading in a superb Lukasz Piszczek cross before Aubameyang grabbed his 22nd league goal of the campaign with a stoppage-time tap-in.
Schalke 04 missed a chance to climb into the top four after a lacklustre 0-0 draw at Eintracht Frankfurt left them in seventh spot.
Borussia Moenchengladbach are fourth despite their 2-2 draw at Augsburg.
Bayer Leverkusen's European hopes suffered a setback with a 3-1 defeat at Mainz that saw them drop to sixth on 35 points, one behind their opponents.
Hertha Berlin are third on 39 points following Friday's 1-0 win at Cologne.
(Reporting by Karolos Grohmann, editing by Ian Chadband)
