"It was an emotional thing for me to play with him here in this court", Nadal said.
In other matches, Juan Martin Del Potro has eased into the fourth round of the French Open by beating No 31 Albert Ramos-Vinolas 7-5, 6-4, 6-1.
Marin Cilic has moved into the fourth round at a Grand Slam tournament for the 20th time, eliminating Steve Johnson of the U.S. 6-3, 6-2, 6-4 at the French Open.
"I'm disappointed to lose, you always want to do better, but I'm not frustrated", said the Frenchman.
Del Potro next plays either ninth-seeded John Isner, the last American left in the men's singles after eliminating the last Frenchman standing. He's a player that has a good serve, good forehand.
Goffin led by a break in the third set with the score at one set apiece overnight, only for Monfils to roar back and win the third set.More news: Facebook gave 60 device makers "deep access" to user data
Next up for Williams as she competes in a major tournament for the first time in 16 months - and since giving birth in September - will be a familiar foe: Maria Sharapova.
Rafael Nadal has been full of focus at Roland Garros.
Nadal was then untouchable on serve during the next two sets, with his serve never once in danger, while he broke his rival's serve two times in each set.
Meanwhile, South African sixth seed Kevin Anderson saw off Germany's Mischa Zverev 6-1 6-7 (3-7) 6-3 7-6 (7-4) to set up a last-16 meeting with Diego Schwartzman.
There was, however, one upset in the women's event today as 25th seed Anett Kontaveit of Estonia beat two-time Wimbledon champion Petra Kvitova of the Czech Republic 7-6 (8-6), 7-6 (7-4).
The Zverevs could have been just the third pair of brothers to reach the round of 16 at the same major in the 50-year Open era.More news: Justin Trudeau: Donald Trump's comments on Canadian steel industry 'insulting'
David Goffin saved four match points to close out his third-round tussle with Gael Monfils in just short of four hours after it was carried over from Friday.
If it hadn't been for 18-time grand slam singles victor Chris Evert, Dubreuil's dream may never have come true.
The 27th seed insisted he doesn't respect Nadal too much on court, though, pointing out that he's not the first man to struggle against the 10-time French Open champion."It's not a question of respect".
Fognini, sporting an eye-catching red T-shirt with black thunderbolts on the front and back, beat 16th-seeded Briton Kyle Edmund 6-3, 4-6, 3-6, 6-4, 6-4.
Fognini joins compatriot Marco Cecchinato in the fourth round, the first time that more than one Italian has made the last 16 since 1976.More news: Trump criticizes Canada's lumber sector after imposing steel, aluminum tariffs
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