The former junior rivals first played at tour level back in 2004 and Nadal has won every encounter across 18 years.
Gasquet has not won a set since 2008 and for more than an hour even a game was beyond him. He eventually stopped the rot at 6-0 3-0, accepting the crowd’s acclaim, and at least made a fight of it in the third before going down 6-0 6-1 7-5.
After being dealt a blow on the nose with his own racket in his second-round victory over Fabio Fognini, it was smooth sailing this time for Nadal, who was happy with his step up in performance.
“It’s a little bit bigger than usual but it’s OK, the nose is still there,” he said with a smile.
“Three matches already, I increased a little bit the level today so that’s important for me I think. I’m happy for that against a very good friend like Richard. I need to keep playing better and better to have chances to go through.”
Nadal, who still has not lost a grand slam match this year, next plays American Frances Tiafoe, who reached the fourth round for a third straight year with a 7-6 (7) 6-4 6-4 victory over Diego Schwartzman.
Third seed Carlos Alcaraz continued to scythe his way through the draw, seeing off American Jenson Brooksby 6-3 6-3 6-3 on Saturday, while ninth seed Andrey Rublev came through an epic tussle with Denis Shapovalov.
It went all the way to a deciding tie-break, with Rublev finally winning 6-4 2-6 6-7 (3) 6-4 7-6 (7) after four hours and eight minutes.
He next meets seventh seed Cameron Norrie while Alcaraz will play former champion Marin Cilic.