Potapova harnessed her receiving game to dominate the American, winning 6-2, 6-3 to reach the quarter-finals of the clay-court tournament.
World number six Gauff, who made it past Potapova's countrywoman Veronika Kudermetova in three sets despite an error-ridden second-round performance on Wednesday, converted none of her six break points against Potapova.
Potapova, ranked 24th, is competing in Stuttgart for the first time with the 22-year-old advancing to her fourth quarter-final of the year.
Earlier on Thursday, Czech former world number one Karolina Pliskova fought past Croatian Donna Vekic 6-2, 6-7 (5/7), 7-6 (7/5).
Pliskova, who won at Stuttgart in 2018, dominated the first set but lost the second in a tie-break. She closed out victory in the third in another tie-break.