NEW YORK — Her secret to beating Serena Williams two years ago at the U.S. Open was that Karolina Pliskova felt free, which is perhaps the best and most improbable mind-set there is when you’re plunked in the middle of 24,000-seat Arthur Ashe Stadium with a then-22-time Grand Slam champion staring back across the net.

Pliskova had arrived in New York practically unknown except to tennis insiders when she beat Williams in the semifinals at Billie Jean King National Tennis Center in 2016. The 24-year-old had six WTA Tour titles to her name but little Grand Slam success; she began the tournament as the only top 20 player in the draw who hadn’t yet made it to the fourth round of a major tournament. She won with a powerful combination of freedom from the burden of expectation and confidence that came from 10 consecutive victories leading into that match. Her serve helped, too.

“I really was feeling great that year. I’m feeling great now, too, but it was a little bit different story, 2016,” Pliskova said Sunday. “I was the dark horse. Nobody was expecting me to get that far. But I just went on the court just to get the win — it was not for me just to go and enjoy, or just to go and do some games against her.

“That’s why I won, because I believed I had a chance. I have a game to beat her. That’s what I would like to do if she’s there this time, too.”

That day, Pliskova became a member of the illustrious club of 10 women, only three of whom are still active, who have ever defeated Williams at the U.S. Open, joining names such as Clijsters, Henin, Capriati, Davenport and the other Williams.

Because Williams skipped last year’s tournament in New York, Pliskova is also the last player to beat the six-time champion in the U.S. Open.

On Tuesday in the quarterfinals, they will have a rematch. The Czech defeated Australia’s Ashleigh Barty, 6-4, 6-4, and Williams beat Kaia Kanepi, 6-0, 4-6, 6-3, to set it up.

This time, Pliskova won’t be the dark horse. Although she still doesn’t have a Grand Slam title to her name — the 2016 U.S. Open final that she lost to Angelique Kerber is still her best result in a major — she owns 10 WTA titles and is ranked No. 8 in the world. She hasn’t dropped a set yet this tournament.

“It will be, for me, a dream to play again on Arthur Ashe, and to play Serena, because obviously last match we played, I won. Since then, we didn’t play. A lot of things happen between,” Pliskova said. “I just feel like I’m handling those situations much better. I know how it is to be in the final, actually in the second week of Grand Slams. So that’s what makes me confident that I can do it again.”

The Czech will be running into a woman on a roll Tuesday. Williams continued with the nearly flawless tennis she displayed in the third round against her sister Venus in Sunday’s first set against Kanepi, the woman who upset the No. 1 player in the world, Simona Halep, on the first day of the tournament.

Williams’s level dropped in the second set, but she let out a booming “Come on!” after winning the first game in the third and cruised from there. She finished with 47 winners to Kanepi’s 22 and served 18 aces, earning her 10th consecutive quarterfinal spot at Flushing Meadows.

When asked whether Williams was playing as well as the 33-year-old Kanepi had ever seen, the Estonian’s response was simple.

“Yeah,” she said. “[Her serve] today, it was unreturnable.”

Williams’s fellow American John Isner booked his trip to the quarterfinals in the men’s draw Sunday, defeating No. 25 seed Milos Raonic, 3-6, 6-3, 6-4, 3-6, 6-2, in Louis Armstrong Stadium.

In the women’s draw, Anastasija Sevastova advanced to the quarterfinals with a 6-3, 1-6, 6-0 win over Elina Svitolina. Defending champion Sloane Stephens became the fourth woman through to the quarters with a 6-3, 6-3 rout of No. 15 seed Elise Mertens.

Stephens and Williams could be on a collision course for the semifinals, where Stephens beat Venus last year. The 25-year-old showed some of her sharpest, most aggressive tennis of the tournament against Mertens, the 15th seed, who beat her last month at a warmup tournament in Cincinnati.

But first Stephens has to play Sevastova, the same woman she defeated in three sets in the quarterfinals last year.

On Sunday, Williams was looking forward to her own rematch.

In Williams’s previous match against her sister, the 36-year-old displayed a kind of mental toughness tennis fans hadn’t yet seen since Williams returned to the tour after giving birth — the U.S. Open is just her seventh event back.

On Sunday, she showed progress again. The second set against Kanepi is the first Williams has lost in this U.S. Open, and the effort it took to raise her level back to be in shape to win the third set so assertively was impressive. She both made tactical adjustments and simply out-hit the big-swinging Estonian.

Even after her time away, Williams’s power still can’t be matched on court.

“It wasn’t easy, obviously, but I think one thing she did well was change her strategy and started doing just a completely different game and gave me a different look to her game,” Williams said. “I definitely think it was more tactical and, I don’t know, physical, I guess.”

Williams has been watching Pliskova since they last played and noted her impressive forehand and dangerous serve as her greatest weapons.

“She was No.1 last year,” Williams said, “I think she got there for a reason.”