But she couldn't convert two break chances in the 10th, and after Williams saved two set points Kanepi knotted the match with a service victor on her third set point.
She continued: 'For the first time, I finally understand the concept of self-love and self-care. She was bumped up to the No. 17 seed at the U.S. Open.
Every interview she had conducted through the first week of the U.S. Open fortnight, from court-side chats, to post-match news conferences, now come with the obligatory mother/daughter story-line.
On Tuesday in the quarterfinals, they will have a rematch.
"I know she has a big game, but I have a big game, too", Pliskova said after her 6-4, 6-4 win over Australian Ash Barty on Sunday.
This time, Pliskova won't be the dark horse.
Sevastova, in the quarters for a third straight year, is hoping to turn the tables on Stephens, who is trying to become the first woman to win back-to-back US Open titles since Williams won three straight in 2012, 2013 and 2014.
Kanepi turned things around early in the second set, as Williams' serve speed dropped and Kanepi picked up her level. "I just feel like I'm handling those situations much better. Just simple: Make them play an extra ball, and see what happens".
The Czech will be running into a woman on a roll Tuesday. Plus, it's worth remembering this- Kanepi eliminated No. 1 Simona Halep on Day 1 of this tournament.
Yet, Kanepi would have an answer in the second set with a 6-4 win to draw level.
That was all she needed as 18 aces and 47 total winners proved plenty to put her through.
"He'd have so much to prove", she said.
Williams is still going, on to a matchup with eighth-seeded Karolina Pliskova. But after squandering a 2-1 lead in sets, the 11th seed dominated the fifth set to beat 25th-seeded Milos Raonic 3-6, 6-3, 6-4, 3-6, 6-2.
"It was a Serena scream", Williams said with a smile.
Williams took the first set against Kanepi in just 18 minutes.
Nadal faced Georgia's Nikoloz Basilashvili, the 2018 German Open contender and one of his opponents at Roland Garros in 2017, when the former had a relatively smooth path to victory - a 6-0, 6-1, 6-0 win.
On Sunday, she showed progress again. The American would win the next two games, but she couldn't find another break and after fighting off two set points saw Kanepi knot the match on her third chance. She both made tactical adjustments and simply out-hit the big-swinging Estonian. Kanepi was matching Williams' power with booming groundstrokes of her own.
"I made one or two crucial errors and that cost me the second set, at least cost me the first game", Williams told reporters. She has had a great career.
Asked if she was already thinking about a little brother or sister for Olympia, she said not quite yet.