make a run for it.”
But her form was so shocking that Williams ran to the practice courts to work on serves after the match.
Next up for Williams will be fellow American Bethanie Mattek-Sands, who beat compatriot CoCo Vandeweghe 6-2, 6-1.
“I’ll have to play a little better if I want to win,” Williams said.
Nadal was down 5-4 in the first set tie-breaker before battling back against Schwartzman.
“Happy to be through,” Nadal said. “Was a tough battle. I feel lucky to be through.... To take that first set was very important to me.”
Nadal, the 2010 and 2013 US Open champion, improved to 22-1 in his past 23 US Open matches after missing 2012 and 2014 due to injury and finishing as the 2011 runner-up.
Canadian 10th seed Milos Raonic, who could meet Nadal in the fourth round, fired 18 aces in dispatching Spain’s Fernando Verdasco 6-2, 6-4, 6-7 (5/7), 7-6 (7/1).
Defending champion Marin Cilic and seventh seed David Ferrer neared a fourth-round meeting as well.
Croatian ninth seed Cilic fired 19 aces in defeating 139th-ranked Russian qualifier Evgeny Donskoy 6-2, 6-3, 7-5.
“It’s sort of unreal. I was working all my life to win a Grand Slam and now I’m working to defend one,” Cilic said.
Spain’s Ferrer, the 2013 French Open runner-up who missed the past 2 1/2 months with an elbow injury, downed 102nd-ranked Serb Filip Krajinovic 7-5, 7-5, 7-6 (7/4).
“I’m very happy I’m in the third round,” said Ferrer. “It’s a nice comeback playing on these courts.”
Serena Williams could reach a Grand Slam singles final without facing a top-10 rival for the first time in her career, but her quarter-final foe could be 35-year-old sister Venus or Swiss 18-year-old Belinda Bencic, who inflicted Serena’s most recent defeat.
Bencic, the highest seed remaining in Williams’ half of the draw at 12th, saved three match points in the second set and outlasted Japan’s 88th-ranked Misaki Doi 5-7, 7-6 (7/3), 6-3.
“I’m just happy I could turn it around,” she said.
Bencic, who defeated Serena Williams in the Toronto semi-finals last month, next plays Venus after the elder Williams eliminated fellow American Irina Falconi 6-3, 6-7 (2/7), 6-2.
“She played really well in the tie-breaker,” Venus Williams said. “In the third set I had a lot more clarity.”
Australian Open semi-finalist Madison Keys ripped 100th-ranked Czech Tereza Smitkova 6-1, 6-2. The American 19th seed hopes for a fourth-round date with Serena Williams, who ousted her in Australia.
“Fingers crossed it could happen. If it does happen I could come out and have some fun at my home Slam,” Keys said.