Serena Williams celebrates after winning a point Saturday against Sloane Stephens in the third round of the U.S. Open in New York. Williams advanced with a 2-6, 6-2, 6-2 victory. Seth Wenig/Associated Press

NEW YORK — In danger of her earliest exit at the U.S. Open since her debut 22 years ago, Serena Williams turned things around and took over against Sloane Stephens.

Williams emerged from the third-round matchup between two Americans who are past champions at Flushing Meadows – she’s won six of her 23 Grand Slam singles titles at the place – with a 2-6, 6-2, 6-2 victory over 2017 champion Stephens on Saturday.

Afterward, Williams’ 3-year-old daughter, Olympia, wore a mask and waved at Mom while sitting on her father’s lap in a front-row seat. On her way to the locker room, Williams waved back.

“I hope,” Williams said, “that she saw her mama fighting.”

Williams did it, as she so often does, with her best-in-the-game serve, hitting 12 aces at up to 122 mph, and an ability to recalibrate her groundstrokes that were so awry early and so superb down the stretch.

“She served a lot better,” said Stephens, now 1-6 against Williams, although they hadn’t played each other since 2015. “Obviously she has one of the greatest serves in the game. It’s really difficult to read.”

Williams won 10 of the last 12 games by lifting her level, to be sure. But it helped that Stephens went from playing nearly perfectly to missing more and more. Following mistakes, Stephens would look over at her coach, Kamau Murray, or smack her right thigh with her palm so loudly that it echoed through a nearly empty Arthur Ashe Stadium.

“In that first set, I don’t think she made any errors, honestly. She was playing so clean. I said, ‘I don’t want to lose in straight sets,’” said Williams, whose only career U.S. Open loss before the fourth round came in the third round against Irina Spirlea in 1998 at age 16.

Williams, who turns 39 in three weeks, said she told herself Saturday: “OK, Serena, just get a game. Get a game.”

There were, of course, no fans – banned because of the coronavirus pandemic – and so the whole thing was flush with the feel of a practice session, rather than a high-stakes contest on a Grand Slam stage.

There were some other players on hand to watch, including 15th-seeded Maria Sakkari, who sipped on orange juice and ate lunch while on her player suite’s balcony. Sakkari had a vested interest: She faces Williams for a berth in the quarterfinals.

Three other American women are in the fourth round, including No. 2 seed Sofia Kenin, the reigning Australian Open champion, who beat Ons Jabeur, 7-6 (4), 6-3.

But Madison Keys, the runner-up to Stephens in 2017, retired from her third-round match against Alize Cornet, who was leading 7-6 (4), 3-2. Keys, the No. 7 seed, got treatment from a trainer on her upper back and neck after the first set.

Other women into the fourth round: No. 16 Elise Mertens, No. 20 Karolina Muchova and Tsvetana Pironkova, who defeated No. 18 Donna Vekic, 6-4, 6-1.

In men’s action, 2019 runner-up Daniil Medvedev and another semifinalist from a year ago, No. 6 Matteo Berrettini, moved on, as did No. 10 Andrey Rublev, No. 15 Felix Auger-Aliassime, No. 21 Alex de Minaur and Vasek Pospisil, who’s been in the news lately because he teamed up with Novak Djokovic to set up a new association to represent players.

Pospisil eliminated No. 8 Roberto Bautista Agut 7-5, 2-6, 4-6, 6-3, 6-2, and next faces another five-set winner, de Minaur, who defeated No. 11 Karen Khachanov, 6-4, 0-6, 4-6, 6-3, 6-1.

Medvedev’s next opponent is the only remaining American in the men’s draw – Frances Tiafoe.

Tiafoe, ranked 82nd, dominated Marton Fucsovics in a 6-2, 6-3, 6-2 victory.

Medvedev defeated Tiafoe in the first round of the Australian Open in January.

“He’s a tough out, and he’s confident right now,” Tiafoe said. “He’s playing the best tennis of his life. But I feel like when I’m playing like I’m playing right now, I’ve got a chance against anybody in the world. I’ve felt that way for years, and I’m starting to finally put it together again.”

Williams vs. Sakkari will be a rematch of their meeting won last month by Sakkari at the Western & Southern Open, a hard-court tournament played at the U.S. Open site instead of its usual home in Ohio because of the pandemic.

“You know, Serena is Serena,” Sakkari said after advancing earlier Saturday by beating 19-year-old American Amanda Anisimova, 6-3, 6-1. “You have to come up with some great tennis. Otherwise there is no chance against her.”

Stephens did that very thing in the early stages Saturday.

She was playing patient, wait-for-the-right-moment tennis, creating lengthy, complicated points that more often than not ended with Williams blinking first.

After one 15-stroke exchange in the first set culminated with a backhand that flew wide, Williams stared at her racket strings and smacked them with her left palm.

Stephens moved out in front early, breaking to lead 2-1 when Williams handed over five consecutive points from 30-love with five consecutive groundstrokes that missed the mark.

Quickly, that set belonged to Stephens, who held at love when all four points ended with Williams miscues. In all, Williams made 13 unforced errors in set. Stephens? Just five.

The end of that set was met by total silence – not even the awkward, manufactured soundtrack piped in during other matches to simulate ambient noise.

From 2-all in the second, though, Williams finally made inroads. She saved a break point, then broke for the first time when Stephens missed a forehand, part of a stretch in which Williams grabbed 12 of 15 points.

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