On a blustery day in the California desert, Kostyuk got to grips with the conditions to roar through the opening set in just 22 minutes.
The second set looked to be following the same pattern as Kostyuk raced to a 3-0 lead before Potapova managed to win a game.
She then broke Kostyuk to put it back on serve and suddenly Kostyuk found herself in a battle.
"I mean, 6-0, 3-0 is probably the worst score in the world, honestly," said Kostyuk, who served for the match at 5-3 in the second and was broken.
She finally polished it off with one last break of serve, advancing to her first 1000-level semi-final when Potapova put a forehand into the net on her rival's second match point.
"Today I was playing the wind, so it was very tricky, and Anastasia is a very tricky opponent," said Kostyuk, who reached her first Grand Slam quarter-final at the Australian Open this year. "She always fights, so I tried to stay there until the end.
"I feel I got a little bit passive, waiting. She was doing a lot of unforced errors in the beginning of the second set so it was not easy when she was getting back in the game."
Kostyuk will next play either world number one Iga Swiatek or former number one Caroline Wozniacki.
The only previous meeting between Swiatek and Wozniacki came in 2019 - just seven months after Poland's Swiatek had made her WTA main draw debut and five months before Wozniacki retired.
Wozniacki, now a mother of two, returned to the sport last year.
She faces a tough test in Swiatek, who has dropped just 10 games in three matches this week.
In the other women's quarter-finals, US Open champion Coco Gauff, seeded third, takes on China's Yuan Yue while ninth-seeded Maria Sakkari of Greece faces American Emma Navarro.
Both Gauff and Sakkari could be tested. The big-hitting Yuan earned her first title in Austin two weeks ago while Navarro knocked off Australian Open champion Aryna Sabalenka in the fourth round.
