The second-placed Ruhr valley club, who have five games left in the season, maintained their slim title chances as they are on 60 points, six behind Bayern Munich, who host Augsburg on Saturday.
Dortmund, who had lost 4-1 at home against Leipzig last week, suffered some bad luck at the start with Gio Reyna being taken off injured after only two minutes and Brandt coming on.
The Norwegian, a major transfer target, provided the assist but has not scored in his last five league matches, his longest goal drought in the Bundesliga.
With Stuttgart missing in-form striker Sasa Kalajdzic, they were on the backfoot after the break and Brandt struck again in the 71st. He rifled in from the edge of the box to give Dortmund a two-goal cushion.