The teenage winger inadvertently broke the deadlock in the 38th minute at Signal Iduna Park when he swung the ball in from the right and saw it drift over Ron-Robert Zieler and in off the far post.
It was the slice of luck Dortmund needed to overcome a nervy start as they bounced back from last weekend's 6-0 hammering at Bayern Munich.
Michy Batshuayi and Maximilian Philipp both struck after the break to seal the victory, moving BVB within a point of second-placed rivals Schalke ahead of next Sunday's Ruhr derby.
"I was looking for the back post, just for a cross," Pulisic said of his fourth Bundesliga goal this term.
"It came off my foot a little wrong but it wasn't so wrong in the end. It was a bit lucky but I'm happy it went in."
BVB boss Peter Stoger conceded his team needed time to overcome the shock of the Bayern defeat.
"In the first half you saw why Stuttgart is doing so well this year. It was very difficult for us, because the opponent dared to do a lot," he told Sky Sports Germany.
"At half-time we discussed what we needed to change and after the break we implemented many things much better, with which we are very satisfied with.
"It was not easy to deal with the situation after the defeat in Munich. But it was really good in the second half."