NEW YORK — A crackling Saturday night at the U.S. Open wound up notable mostly for a crackerjack display of sportswomanship. First, No. 1 Naomi Osaka’s 6-3, 6-0 win over both 15-year-old sensation Coco Gauff and the Cocomania sweeping these tennis premises proved pretty much a formality.

Next came an informality laced with grace, when Osaka asked Gauff to refrain from the loser’s customary quick exit and remain on court alongside Osaka for the interview normally reserved for the winner alone. Gauff, still 15 and not even yet 15½ , demurred at first and feared crying in front of an audience. Osaka, 21, kept asking and suggested Gauff postpone her cry until the shower. Gauff relented and closed her remarks as rapidly as she could with, “I don’t want to take this moment away from her because she really deserves it.”

“No, I mean, it was kind of instinct because when I shook her hand I saw she was kind of tearing up a little, then it reminded me how young she is,” Osaka said. She said she figured that “normal people don’t watch the press conferences unless they’re fan-fans,” and so, “I was thinking it would be nice for her to address the people who watched her play.” And: “For me, I just thought about what I wanted her to feel leaving the court. I wanted her to have her head high and not walk off sad.” And: “I feel like the amount of media on her now is kind of insane, so I just want her to take care of herself.”

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“She just proved that she’s a true athlete,” Gauff said. “For me, the definition of an athlete is someone who on the court treats you like your worst enemy but off the court can be your best friend.”

As the good feeling swirled in a stadium with two tennis families who know each other and whose fathers speak often, Osaka acknowledged Gauff’s parents and teared up. Gauff’s parents beamed at Osaka. Osaka ­finished, Gauff exited, and Osaka applauded her.

Said Gauff: “She was crying; she won. I was crying. Everybody was crying. . . . I don’t know why she was crying. I was like, ‘You won the match!’ ”

That, she did. If anything, the 65 minutes displayed the know-how the Japanese-Haitian raised in the United States has attained in her ­marches to the 2018 U.S. Open title, the 2019 Australian Open title and two turns at the No. 1 ranking. Her groundstrokes proved more durable, seared more and wreaked more gasps.

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“She trusts her strokes a lot,” Gauff said, “so that’s why she hits winners. In order to hit a winner, you have to trust that you’re going to do it. I can work on that more.” Osaka whacked 24 winners to eight for Gauff and made 17 unforced errors to Gauff’s 24, yet she trimmed those errors from 12 in the first set to five in the second. She won the ace category by 5-2 and (happily) lost the double-fault department by 7-1.

As they began amid fanfare and a crowd pulling hard for the teenager, Osaka won all three first games at 30 and posted a 7-1 lead in winners to accompany her 3-0 lead all told. Gauff held serve for 3-1 with a 105-mph ace up the middle, a 119-mph ace up the middle and some hand-to-strings applause from Osaka.

Gauff never held serve again, going 1 for 7 in her service games. She won 42 percent of first-serve points to 74 percent for Osaka. Thus did her U.S. Open conclude along a similar impressive track as her breakthrough Wimbledon, her road ending when running up against the upgrade in caliber present in a top-10 player. At Wimbledon, it was the seventh-ranked player and eventual champion (Simona Halep, by 6-3, 6-3), and here, it was a No. 1 who has reached her first Grand Slam fourth round since her title in Australia.

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It was also an occasion that brought, to the front rows at opposite ends of the court, two sets of parents with remarkable story lines.

Osaka was born in Japan to a Japanese mother, Tamaki Osaka, and a Haitian father, Leonard Francois, who met in Hokkaido. The parents soon moved Naomi to Long Island, and for a blueprint of how to develop two daughters, Naomi and older sister Mari (who has reached No. 280 in the world), Francois had Richard Williams, the father of Venus and Serena Williams.

Gauff was raised in Florida by a former Florida State women’s track heptathlete and a former Georgia State men’s basketball player. Candi, her mother, was a five-time Florida high school champion and a Junior Olympic champion. Corey, her father, played on a Georgia State team that beat Stetson, Texas San ­Antonio and Little Rock to win its 1991 conference tournament. Then Mr. Gauff scored six points with five assists and three steals in a 117-76 loss to Nolan Richardson’s powerhouse Arkansas in the first round of the NCAA tournament.

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Tennis has brought the two families into acquaintance, just as New York, that bastion of international meetings, brought them into Saturday night. Osaka and Gauff speak of each other amiably, and Gauff said: “I mean, my dad and her dad have known each other for a long time. They always talk all the time. We’ve known each other for quite a few times now. She’s a great person. She’s nice. Her mom is one of the nicest people ever. She’s so amazing. Her whole family is just great.”

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