Gauff, 18, was a toddler when Estonian Kanepi -- who at 36 was the oldest remaining woman -- played the first of her two French Open quarterfinals in 2008. On Friday, Gauff had six aces and five double faults, while Kanepi made 29 unforced errors.
"I knew it was going be a tough match. [She] beat [former champion Garbine] Muguruza in the first round. ... I knew today would be close," said Gauff, who achieved her best Grand Slam result when she made it to the French Open quarterfinals last year. "I've been coming to France since I was 10 and trained at the Mouratoglou academy, so I guess it makes me maybe not a clay-court expert, but not bad at it."
Kanepi broke in the first game, but Gauff struck back emphatically, winning the next five games with a flurry of winners on Court Suzanne Lenglen. Kanepi took a 3-2 lead in the second set, but her hopes of a comeback were short-lived, as she dropped serve again when she buried an easy backhand into the net after a poorly executed drop shot from Gauff.
Serving at 4-3, Gauff held and closed out the match on serve when Kanepi made her 29th unforced error.
In other results, Angelique Kerber's quest of completing a career Grand Slam came to an end after the 21st seed lost 6-4, 7-6 (5) to Aliaksandra Sasnovich. Sasnovich will head to the fourth round at the clay-court major for the first time.
Stephens arrived in Paris on a five-match losing streak, but she is 3-0 on this trip.
"I don't think you ever know when it's going to happen or when it's going to click, but I'm just trying to make the most of it, honestly," said Stephens, the 2018 runner-up at Roland Garros. "Happy to put the wins together now. Didn't at any other tournament, so God bless."
Teichmann amassed 53 winners and 58 unforced errors in her win over the 15th-seeded Azarenka.
The 17th-seeded Fernandez broke Bencic in the 11th game of the deciding set and served out the match. She had saved two set points in the opener.
The 27th-seeded Anisimova, an American who reached the Roland Garros semifinals at age 17 in 2019 and eliminated four-time Grand Slam champion Naomi Osaka in the first round this year, was leading 6-7 (7), 6-2, 3-0 when Muchova walked to the net to concede the match.
Muchova twisted her right foot while serving at 2-all in the second set. She went to the sideline during that game to get checked on by a doctor and a trainer, who taped up the ankle.
When play resumed after the medical timeout, Muchova dropped seven games in a row before retiring from the match.
The Associated Press and Reuters contributed to this report.