The win marks the fourth time this year Potapova has beaten a player ranked in the top 10.
"It was my patience, just to stay focused as much as I could because Ons is very tough to play," the 22-year-old Russian said.
"She's truly one of the best in the world. With her style, it's very uncomfortable, I'm really happy and proud to get this win."
World No.7 Jabeur landed just 39 of her 85 first-serves, had nine double-faults and struggled to find any rhythm.
While the win was sweet for Potapova, she said it may have come at the cost of family relationships.
"Tell me about how popular she is - my mother, she loves her so much," Potapova said of her well-liked Tunisian rival.
"Every time I talk to my mum on FaceTime with Ons around, she's like, 'Call Ons. She's my best player'.
"They're actually pretty good friends. I don't know if she's happy that I won today."
Third seed Maria Sakkari of Greece advanced when Colombian qualifier Camila Osorio bowed out due to a right thigh injury while trailing 6-3 2-2.
"It's always very sad and unfortunate to see any athlete, any player, being injured and not being able to compete at 100 per cent," Sakkari said.
France's Caroline Garcia avenged a three-set loss to Sloane Stephens at last month's Cincinnati Open, advancing to the last-eight with a 6-3 3-6 6-1 win.
"Today was a revenge day," said Garcia, who converted four of her seven break-point opportunities.
"The beginning of the third set was really tight. I faced a break point in the first game and managed to stay in front. That was super-important.
"The last couple of months wasn't always great tennis, but we try to fight, we try to train hard."
Czech Barbora Krejcikova, the fourth seed, overcame eight double faults to oust Ukraine's Anhelina Kalinina 6-3 6-2.
"It was really a difficult first set, even though the score doesn't look like that," the 2021 French Open winner said.
"I wouldn't really say I was in control after that, but I was just trying to hit my shots and stay aggressive."
American wild cards Sofa Kenin and Katie Volynets faced off, with Kenin recovering from a shaky start to win 1-6 6-4 6-2, while another US player, Danielle Collins, was a 2-6 6-3 6-4 winner over Latvia's Jelena Ostapenko.
