The world number one won his 29nd straight set at Roland Garros since withdrawing from a third-round match with Marcel Granollers in 2016 due to a wrist injury.
He is only three behind his own haul of 32 straight sets won between 2007-2009, nine behind the record mark of six-times winner Bjorn Borg of 41 between 1979 and 1981.
The 10-times champion had far too much class and power for 27th seed Gasquet - who he remains unbeaten against in 16 professional meetings - in what already appears to be a formidable march to an 11th French Open title in Paris.
Nadal gives ball boy dream chance of Roland Garros knockabout
He will face Germany's Maximilian Marterer - conqueror of Denis Shapovalov in the last 64 - on Monday for a place in the quarter-finals.
Top seed Nadal won the first 12 points of the match and opened up a 5-0 led before Gasquet finally got into the contest, but it was once more too little, too late for a player who had been labelled the 'Mozart of tennis' in his teenage years.
Gasquet actually won their last encounter - but it was at the Kooyong Classic exhibition event this year.
When it matters Nadal must know he won't get into trouble, having not dropped a set against a player who was his rival in the juniors since 2008.
Nadal produces vintage form to thump Gasquet
There was a sense of inevitability about the outcome when both players made their appearance on Court Philippe Chatrier, where Nadal is the huge favourite to lift the Musketeers Cup for a record-extending 11th time.
"I played very well, I'm sorry for Richard, he is a good friend and a good person," said Nadal, who played a few rallies with a ball boy after his match.
He has now won 30 sets in a row on the Parisian red dust and his last-16 opponent, 70th ranked 22-year-old German Marterer, does not exactly seem equipped to cope with the Mallorcan's range of weapons.
Gasquet was totally overwhelmed on Saturday, finding himself trailing 5-0 after 12 minutes, and the Parisian crowd felt it could be an express washout.
Nadal: Fast start was key to victory
In a bid to give the home fans something to shout about, Gasquet spurred himself into action and with a few beautiful shots reduced the gap to 5-3.
But Nadal was in no mood for a marathon and he bagged the first set on serve before putting Gasquet to the sword again at the beginning of the second and third sets, making sure there was no hope of a comeback amongst the Frenchman's supporters.