Gauff, the No. 6 seed, held all of her serves in the opening set and finished the 45-minute session with three aces. Osaka had the first double fault of the match early in the second set and Gauff took advantage for a 2-1 lead. Gauff quickly built a double-break lead at 5-1 and led 40-0 before Osaka saved four match points and used her first ace to make it 5-2.
Osaka then broke Gauff for the first time in the match and overcame another 40-0 deficit with five straight points for 5-4. But Gauff closed it out on her eighth match point with an unreturnable serve.
"The most proud was how I responded mentally," Gauff said. "I could have easily folded in those couple of double faults and match points that I lost. But I kept telling myself, 'Another point, another chance,' and then the music kind of helped on the bench. I was trying to sing along to the song to get me out of the match."
Gauff's victory evened the series with Osaka at two wins apiece. Gauff will face No. 2 seed Paula Badosa on Friday.
Rogers beat top-seeded Maria Sakkari 6-1, 6-3 to reach the quarterfinals for the third time this season. Coming off an opening-round win against 2019 US Open champion Bianca Andreescu, she beat Sakkari, a 2018 finalist and 2019 semifinalist in San Jose, for the third straight time.
Sabalenka came back from trailing 5-3 in the third set to edge Caroline Dolehide 5-7, 6-1, 7-5. Sabalenka overcame 20 double faults in the match to advance to play No. 7 seed Daria Kasatkina of Russia.