The ninth seed relied on his trademark backhand to unsettle Kyrgios, who appeared to be suffering from shoulder pains and needed treatment from the trainer in the opening set.
Gasquet, who has yet to drop a set, played neatly and although he was at times on the back foot because of his opponent's booming forehand, 17th seed Kyrgios's 44 unforced errors were enough to see the Frenchman through.
The world number 10, who next faces Japanese fifth seed Kei Nishikori, had not lost a set in two previous claycourt matches against Kyrgios and it quickly became obvious the Australian would struggle.
"This is the best format for me. His service was less painful here than on other surfaces," Gasquet, who will be playing the fourth round at Roland Garros for the fifth time in six years, told reporters.
"I was able to find the right angles. This being said, it was important for me to win the second set. I knew that one set each it would have been more difficult."
To Kyrgios, it was a no contest.
"I got absolutely destroyed. Wasn't really fun," the 21-year-old said.
Gasquet now has to prove he has the engine to finally go further at his home slam, having never reached the quarter-finals at Roland Garros.
He has a 6-2 win/loss record against Nishikori but has lost their last two matches in straight sets on clay in Madrid and Rome this year.
"I know what to expect against him," said Gasquet.
"He plays very fast. He's a very talented guy, so he takes the ball very high, so one of the toughest players to play for me now. It's different, it's five-sets match in Paris. I want to give 100 per cent and we will see."
(Editing by Ed Osmond)
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