The eight-time champion lost just five points in a 16-minute first set but faced break points for the first time in this year's tournament, saving all four. Federer has now won 32 consecutive sets at Wimbledon to move within one straight-sets victory of breaking his previous longest streak at the All England Club, when he won 34 in a row between the third round in 2005 and the final in 2006.
Later on Monday, Rafael Nadal also extended his streak of not dropping a set in Wimbledon. The second seed defeated Jiri Vesely 6-3, 6-3, 6-4 on Centre Court. Nadal, a two-time Wimbledon champion, reached his first Wimbledon quarterfinals since 2011, when he was runner-up to Novak Djokovic in 2011.
When asked whether he would rather not play Federer in the final, should they meet, Nadal said, "If he's in the final, I am excited to play Roger. I am not saying I am not excited to play against Roger in the final. ... Facing Roger again [would] be something fantastic.
"If you ask me if I prefer another one, I say yes. That's the point. It's about being smart, no?"
Djokovic, seeded 12th, won easily over Russia's Karen Khachanov, 6-4, 6-2, 6-2. Djokovic reached his 41st career major quarterfinal, tying Jimmy Connors for second all time.
It's the first time Isner, 33, has reached the Wimbledon quarterfinals, and it is just his second quarterfinal at a major. His only other quarterfinals appearance was at the 2011 US Open.
The win also allowed Isner, the highest-ranked U.S. man, to have something about Wimbledon to relish other than his victory in the longest match in tennis history: 70-68 in the fifth set against Nicolas Mahut after 11 hours spread over three days in 2010.
He'd never even been past the third round at the grass-court Grand Slam until this year.
"Certainly this tournament, since that long match, has sort of been a house of horrors for me," Isner said. "I've lost a lot of close ones since that match in 2010 -- a lot of very, very close ones, a lot of deep five-set matches, third round especially."
Kevin Anderson also advanced to his first Wimbledon quarterfinal. He edged Gael Monfils 7-6 (4), 7-6 (2), 5-7, 7-6 (4), to whom he held a 0-5 head-to-head record entering the match. Anderson, who reached his first Grand Slam final at the 2017 US Open, had lost in the fourth round at Wimbledon on three previous occasions.
The eighth-seeded Anderson will meet defending champion Federer in the next round.
No. 24 Kei Nishikori avoided an upset, winning 4-6, 7-5 (5), 7-6 (10), 6-1 over unseeded Ernests Gulbis of Latvia. He will face Djokovic next in his first quarterfinals appearance at the All England Club.
Information from The Associated Press was used in this report.