Ranked at number 219 in the world, Stollar stunned the sixth seed with 28 winners in an hour and 20 minutes to seal her fourth successive victory on the clay in South Carolina.
Konta, the world number 22, received a bye in the first round but could only muster seven games in the match as Stollar avenged her defeat to the Brit in the Fed Cup back in February.
"I think I pushed my game to her and I stayed really aggressive and I could push her back behind the baseline, so that helped a lot," Stollar said.
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"I'm going to just try to play my game and just keep being aggressive. It feels good, you know, but it's still not over. I still have matches to go and I don't want to stop here."
The 19-year-old will next take on 12th seed Kiki Bertens, who beat Serbia's Aleksandra Krunic 6-4 6-2.
Meanwhile, Kristyna Pliskova rallied from a set down to beat second seed Petra Kvitova 1-6 6-1 6-3 in an all-Czech clash.
Kvitova, a two-time Wimbledon winner, won a lopsided opening set before her 77th ranked opponent took control.
Pliskova's aggression saw her cruise through the second set and she went on to break her opponent twice in the decider to book a third round spot in the WTA's first clay court event of the season.
"My serve was horrible in the first set, so I had to focus on my serve and the rest of the game was better then as well," Pliskova said.
"I'm really happy, it's my second top 10 win this year, but also it's against Petra."
Last year's US Open finalist Madison Keys defeated Lara Arruabarrena of Spain 6-1 3-6 6-3 and will face Italian Camila Giorgi in the third round.
Additional reporting from Reuters
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