Marin Cilic, who won the 2014 U.S. Open, lost in the third round.
At 19, Nadal said, "you have a lot more years to enjoy the tour, a lot more years to enjoy the life". Federer then narrowed it to two Monday night with a straight-sets win over Phillip Kohlschreiber.
He never gave the unseeded Dolgopolov a chance, breaking serve six times in the 1-hour-41-minute match. Nadal has gotten better and better as the tournament has progressed.
Rublev practiced with Nadal in Monte-Carlo a few years ago. Rublev knocked out No. 9 seed David Goffin on Monday and before that took out Grigor Dimitrov, the seventh seed.
Federer is making his 12th U.S. Open quarterfinal appearance, which is tied with Ivan Lendl for third-most such appearances by a man in the Open Era - trailing Jimmy Connors' 17 and Andre Agassi's 13, according to ESPN Stats & Information research.
The last match of the night saw Madison Keys beat Elina Svitolina in three sets, 7-6 (2) 1-6 6-4, to become the fourth USA player to reach the quarter finals - along with Venus Williams, CoCo Vandeweghe and Sloane Stephens.
"I'm gonna repeat", Nadal said.
After having dropped the opening set of his last two matches, the world number one got off to a far better start in his fourth-round match, although he had some help from his opponent. If I am here with victory and I will answer you with a lot of great pleasure if that happen. Now I have Rublev in front. "I mean, I have incredible respect for him". "I played a very clean match".
Asked about a potential first-ever meeting with Federer at the US Open in the semi-finals next week, he said: "I am in the fourth round now".
The world number one brushed aside world number 64 Alexandr Dolgopolov 6-2 6-4 6-1 in just an hour and 42 minutes on the Arthur Ashe Stadium. She won 25 points to just seven for Brady in the second set. Thiem served for the match at 5-3, but got broken.
The Ukranian then upset Tomas Berdych, coming back from a set down to win it in four.
Next up for Pliskova is 20th-seeded American CoCo Vandeweghe after she beat Czech Lucie Safarova 6-4, 7-6.
Thiem next faces 2009 champion Juan Martin del Potro of Argentina, who put out Spain's Roberto Bautista Agut 6-3, 6-3, 6-4.
If Federer's back isn't 100 percent, he could have his hands full with del Potro. Not many balls surprise me.
The 19-year-old Russian has played lights-out tennis at this US Open.
Unlike Nadal, Federer made a decision to look ahead and comment on a possible semifinal match against his long-time rival. Rafael Nadal and Roger Federer play their fourth-round matches, and another former champion is in action in a match where he considers himself the underdog. "We both play differently".