The Japanese star, who lost to Cilic in his only grand slam final appearance to date four years ago, gained some semblance of revenge as he triumphed 2-6, 6-4, 7-6 (5), 4-6, 6-4 in a match that lasted four hours and eight minutes.
Tsurenko got his first break points of the match at 4-1 in the second set but Osaka saved all three to put herself on the brink of the semi-finals.
Madison Keys of the United States of America celebrates match point against Carla Suarez Navarro of Spain in a quarterfinal match on day ten of the 2018 U.S. Open tennis tournament at USTA Billie Jean King National Tennis Center.
The 20-year-old Osaka, who has looked comfortable on one of the sport's biggest stages, was tearful after her fourth-round clash but much less emotional after her latest win.
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Williams' only stumble came at the beginning of the match, when Sevastova broke her first service game to win the first two games of the match.
The 29-year-old is in his third semi-final at the US Open and sixth at the majors.
Suarez Navarro enjoyed a fine run in NY, where she upstaged sixth seed Caroline Garcia and ended Maria Sharapova's impeccable record under the lights in the Big Apple, however, the 30th seed could not match Keys.
Osaka's first serve percentage was only 59 percent but she did not concede a point behind her first serve the entire first set and only surrendered seven points on serve over the match. She's still in search of major title No. 1. Part of this has to do with her style of play.
Osaka, the daughter of a Japanese mother and Haitian father, said she enjoyed playing in the blistering heat of NY which has been such a talking point of the tournament.
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"I hate matches like this", said Tsurenko.
"It means a lot", she said.
"I was, alongside many other people, anticipating the match against Federer", Djokovic said. I'm so grateful for the opportunity to play on this court.
Djokovic needed two four-setters in his opening rounds but cruised through in straight sets against Richard Gasquet and John Millman.
There were other issues for him, too, including in the third set when, ahead by a break, he was called by the chair umpire for allowing the 25-second serve clock, making its Grand Slam debut at this tournament, to run out on back-to-back points.
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