Nadal, 18, showed no mercy as he defeated his rival 6-4 6-3 6-2 in less than two hours to progress to the last 16.
Gasquet failed to unsettle Nadal’s physical and mental strength and solid backcourt play in a one-sided contest.
Nadal will next face another Frenchman, Sebastien Grosjean, who needed five sets to overcome Czech Radek Stepanek 6-1 4-6 3-6 6-3 6-4.
Earlier, world number one Federer outclassed Chilean opponent Fernando Gonzalez 7-6 (11/9) 7-5 6-2 to progress to the fourth round.
Roland Garros is the only Grand Slam the 23-year-old Swiss star has not yet won.
Federer will next face 1998 French Open champion Carlos Moya of Spain who triumphed over compatriot Fernando Vicente 6-4 7-6(4) 6-7(3) 0-6 6-4 in a five-set thriller.
Former world number one Venus Williams was stunned after she bowed out in the third round at the hands of 15-year-old unseeded Bulgarian Sesil Karatantcheva.
After losing the first set 6-3, the 24-year-old, who has won four Grand Slams, looked to have regained her composure when the breezed the second set 6-1.
But the Bulgarian teenager, ranked 98th in the world, dominated in the final set to take the match 6-3 1-6 6-1.
World number one Lindsay Davenport felt lucky to still be around after having to come through another three-set battle against France’s Virginie Razzano to progress to the last 16 .
Davenport, for whom clay is far from her favourite surface, is enjoying her campaign.
Davenport will next face Belgian 14th seed Kim Clijsters for the first real battle of the women’s draw .
Clijsters, who reached the French Open final in 2001 and 2003, stormed past 20th seed Daniela Hantuchova of Slovakia in straight sets 6-4 6-2.
Davenport claims the Belgian will be the favourite as they clash on clay for the first time ever tomorrow.
Hantuchova praised the Belgian’s qualities and claimed Clijsters is probably the best female player in the tournament, even though she had a nightmare 2004 season due to a wrist operation.
Fourth seed Elena Dementieva, who reached the French Open final last year, needed three sets to overcome Japanese player Akiko Morigami 6-3 4-6 6-3.
Mary Pierce of France also booked her ticket to the next round after beating Russian ninth seed Vera Zvonareva 7-6 (7/2) 7-5.