The 134th-ranked Vickery created 19 break points but could only convert seven of them against the the US Open champion in Wednesday's round-of-16 clash.
World No.3 Gauff had 15 double faults, but rallied from 4-2 down in the final set before winning in two hours 26 minutes with her first match point.
Gauff, who received a first-round bye at the clay-court tournament, awaits the winner between No.7 Zheng Qinwen and Marta Kostyuk.
Kostyuk earlier overcame former champion Laura Siegemund 6-3 6-7 (7-4) 6-4 in their first-round match.
The Ukrainian player twice served for the match in the second set only for Siegemund, who won the Stuttgart tournament in 2017, to take it to the third.
In Wednesday's other last-16 match, second-seeded Aryna Sabalenka advanced when her opponent and close friend Paula Badosa retired at 3-3 in the third set with a leg injury.
Badosa had a medical timeout while leading 5-4 in the second set and returned with strapping on her left leg.
"I feel so bad for her … I don't have any emotions right now, I'm not happy, I'm not sad," said Australian Open champion Sabalenka, who advanced 7-6 (7-4) 4-6 3-3 retired.
Emma Raducanu powered to a 6-2 6-1 win over former top-ranked player Angelique Kerber in a first-round meeting of two players with wild cards for the draw.
The 2021 US Open champion will meet Linda Noskova in the second round.
Ons Jabeur also had to fight hard to prevail in a 2-6 6-3 7-6 (7-1) win over Ekaterina Alexandrova of Russia.
Jabeur was down a break in the second and third sets before winning.
The Tunisian player will face Italy's Jasmine Paolini in the second round.
Eight of the top-10 ranked women are in action in Stuttgart, the main event in the first week of clay tournaments on the WTA Tour this season.
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