Wimbledon champion Rybakina, fresh off her Indian Wells triumph last week, calmly saved a match point on serve in the second set and fired 60 winners, including a dozen aces, as she earned a gutsy 3-6 7-5 6-3 win.
Rybakina, who also had to go the distance in her opening match against Anna Kalinskaya, struggled to find her rhythm on serve in the opening set but wrestled back momentum in the second.
After being broken four times in the first set, Rybakina held her serve in all but one game in the remainder of the contest and confidently closed out victory in two hours and 31 minutes.
"It didn't start well in the first set but in the end I just found some energy," Rybakina said. "I was trying to push more on the serve and started to move better on the court. It was a really tough battle.
"I was trying to focus on every point because the first set didn't go my way. But it was just a few mistakes here and there. Overall, I played well. Paula was serving well. Just happy that I was able to manage in three sets."
Potapova, seeded 26th, mounted a stout defence to fend off 10 of 13 break points and grinned widely as she fired down an unreturnable serve to secure the win and set up a meeting with China's Qinwen Zheng (20).
"I had this belief in me that I can do this, that I just wanted to fight through and just to show my best tennis and not focus on the score," she told the Tennis Channel.
"It doesn't matter what the score is, you just try to show your best and that's what I did."
Up 5-2 in the first set, it looked as though it might be an easy day for Potapova. But world number six Gauff turned the tables when she converted break point chances in the ninth and 11th games and closed the tie-break with a lethal backhand.
Potapova regained her composure in the second set and pummelled her opponent with her mighty forehand as she broke serve in the penultimate game.
Gauff reached the quarter-final at Indian Wells earlier this month but could not find her spark in the third set.
She failed to earn a single break point as Potapova dropped only one first-serve point and won the final five games.
"In the third set I felt (in control) a lot," said Potapova, who triumphed in Linz last month.
Earlier, the home fans leapt to their feet as American Mackenzie McDonald (27) survived a barrage of aces to upset Italian 19th seed Matteo Berrettini (26) 7-6(8) 7-6(5). He will next face France's Quentin Halys (26).
