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Dortmund was forced into an early change when Gio Reyna pulled up with an ankle problem after barely a minute, making way in tears for Brandt, who had to complete a faster than expected warm-up.
He was alert enough to give Dortmund the lead after just 12 minutes, finding himself in the right place to turn Erling Haaland's pass into an open goal to give the visitors the lead.
Marco Reus almost added a second, bringing a fine save out of Florian Muller, before Stuttgart started to find its feet.
Omar Marmoush dinked an effort over Gregor Kobel, but also the crossbar, as the hosts stepped up their search for an equaliser.
Marco Rose then had to rejig things again as a second Dortmund injury threatened to derail his side, Mahmoud Dahoud leaving the match 10 minutes before the break with a shoulder injury, to be replaced by Axel Witsel.
Stuttgart started the second half strongly, Tiago Tomas almost bagging an equaliser, but he fired wide with the goal at his mercy.
Pascal Stenzel then missed the target by inches when his shot from outside the area just cleared the crossbar.
That profligacy would prove costly as Reus teed up Brandt to take a dip, and he showed his ruthless streak to beat Muller, allowing Dortmund to exhale as the pressure valve was released slightly.
Stuttgart, though, wouldn't lie down. Borna Sosa forced Kobel in a remarkable save with 10 minutes to go, and the Dortmund goalkeeper was thankful for his crossbar when Sosa fired another effort in anger.
But the hosts couldn't find a way, and remain in the thick of a relegation battle, while for Dortmund, victory puts it back within six points of leader Bayern Munich, which will play Augsburg later on Saturday (AEST).