Nadal, the 13-time champion in Paris and chasing a record-setting 21st Grand Slam title, won 6-0, 7-5, 6-2 to register his 17th career victory in 17 meetings with France's 53rd-ranked Gasquet.
"Well, of course it's better to win 16 in a row than lose 16 in a row, no?," said Nadal.
"When you go to the match and you have been winning all the last 16 times, of course the confidence is a little bit higher than when you have a tougher head-to-head.
"But I respect Richard and I know how good he is."
In a match played in an empty Court Philippe Chatrier due to a 9pm Covid-19 curfew, Gasquet won just nine points in the first set.
He managed a brief show of defiance from 2-5 to 5-5 in the second set but eventually his resistance was broken when Nadal carved out a break for 4-2 in the third set.
Next up for the world number three is Britain's Cameron Norrie who he has beaten twice already this season, at the Australian Open and on clay in Barcelona.
Gasquet's defeat means that there are no French players left in the tournament before the third round for the first time.
"It's the end of an era. I'm almost 35. (Jo-Wilfried) Tsonga is 36, (Gilles) Simon is 37, and Gael is (almost) 35," said Gasquet, a former world number seven.
"So it's normal. We are still here. It's a great generation. I hope it will be the same for the French future."