Nadal eased through the second set and made the ideal start to the third, breaking in the opening game after another big forehand.
"I'm happy the way that I was able to win tonight", Nadal said.
She'd just dropped the second set, moments after wasting a match point, as her U.S. Open title defense got off to a shaky start Tuesday.
But the 15-year-old star-in-the-making boasts another quality - one that she gleaned from her grandfather, a former minor league baseball player in the South during the 1960's.
The 21-year-old from Japan wore a black sleeve over her left knee, which has been an issue recently. It was an erratic performance for Osaka, who committed an unsightly 50 unforced errors, 28 more than her opponent, but also blasted 44 winners and pounded eight aces. She looks up to her father the most.
Alexander Zverev, seeded sixth, outlasted Moldova's Radu Albot 6-1, 6-3, 3-6, 4-6, 6-2, while Marin Cilic, the 2014 champion, advanced in straight sets over Slovakia's Martin Klizan.
The Japanese will next face Poland's Magda Linette, who lifted her first career WTA trophy at the Bronx Open last weekend.
But most players aren't Gauff, a 15-year-old from Delray Beach, Fla., who somehow failed to get the memo that being far less experienced than even an 18-year-old opponent is supposed to be a disadvantage. And I could see how stressed out she was when I was playing her.
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Osaka was grateful for their attendance, she said, and motivated to win quickly so she wouldn't keep them in the sun too long watching. Gauff excitedly responded that Obama was an "inspiration" to her.
Gauff, who is based in Florida, double-faulted three times in her first service game.
The Swiss, a five-time victor in New York City, lost the opening set 6-4 before recovering to win the next three sets 6-1, 6-2, 6-4. Gauff began to play steady points of moderate pace, then fantastic points of every pace as she claimed 10 of 12 games while she turned a 3-6, 0-1 deficit into a 3-6, 6-2, 4-1 lead.
And the U.S. Open has some catching up to do.
Gauff, an imposing 5 feet 11, was a two-time junior Grand Slam victor and at 13 was the youngest U.S. Open junior women's singles finalist two years ago.
Venus Williams was not picked to continue on the council after being there "pretty much nearly every year of my career", she said.
"He showed a year ago what he is able to do when he is playing well", he said.
The tour added that what Steiner said violated rules that prevent tennis officials "from discussing specific incidents or matches, individual players, other officials, or rules, in the interest of maintaining impartiality at all times". Along with defending champion Novak Djokovic the "Big Three" have combined to win the past 11 majors and are again strong favourites.