Federer and Williams were making their first appearance at the tournament since they each won the title in 2015. Williams opened with a straight-set victory against Daria Gavrilova.
Federer seeded second, has won in Cincinnati a record seven times but hasn't played there since winning in 2015. In 2016 he had a knee injury, and in 2017 he had back problems.
Nadal will now look to bounce back in the Rogers Cup, a build-up to the US Open where he will be defending his title.
Nadal said it was "an exceptional circumstance" that he has remained on top of the tennis world after turning 30, alongside Federer, 36, and Djokovic, 31.
That trio has combined to win 10 of the past 14 championships at Flushing Meadows: Five for Federer, three for Nadal, two for Djokovic.
Federer had a match point at 5-4 in the third set, but struck a backhand into the net before the eighth-seeded Anderson broke in the following game and held serve.
Federer was seen playing a cricket shot - a near perfect forward defensive push - during the match during his 4th round match.
Roger Federer, Serena Williams and Rafael Nadal have reached the quarter-finals of Wimbledon.
Williams reached her 13th Wimbledon quarter-final by beating Russian qualifier Evgeniya Rodina 6-4, 6-2 on Monday.
Roger Federer moved into the third round at the All England Club as he chases his record-extending ninth championship there.
Roger Federer and Serena Williams are making it look easy at Wimbledon — again. Caroline Wozniacki is heading home early — again.
Lopez, who has reached the Wimbledon quarterfinals on three occasions, took the first set 6-3 against Federico Delbonis of Argentina on Court 7.
The All England Club announced Murray's withdrawal a day before the grass-court Grand Slam tournament begins.
Federer and Nadal will resume their famed rivalry at Wimbledon.
Rafael Nadal returned to the top spot in the ATP world singles rankings released on Monday while Switzerland's Roger Federer came in second.
Federer, who skipped the entire clay-court season for the second year in a row, had been going for his 99th title, one week after taking his 18th grass-court title in Stuttgart.
Federer fired 12 aces and saved six of the seven break points he faced to end Kudla's 10-set winning streak in Halle.
After winning a record-extending 11th French Open title earlier this month, the Spaniard withdrew from the Queen's Club Championships, a key Wimbledon tune-up played in London from June 18 to June 24.
Federer, who won his 18th grass-court title in Stuttgart, is just three match wins away from matching Jimmy Connors' all-time record of 174 victories on grass.
Federer extended his grass-court winning streak to 18 matches, including his titles in Stuttgart last week and at Halle and Wimbledon last year.
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