The 20-year-old Moutet, who has recently trained with Roger Federer and spent time at Rafa Nadal's academy, pulled off an unlikely 6-3 6-1 2-6 7-5 victory over Argentine Guido Pella, the 19th seed who has won more claycourt matches than anyone else in 2019.
Earlier, Tsonga was knocked out by Kei Nishikori and Gasquet, whom Moutet would have faced in the third round, was eliminated by Argentine Juan Ignacio Londero.
"It's fantastic to be in the third round of Roland Garros, of course. And it's also a good sign," said Moutet.
Moutet is now carrying the French flag in Paris, but dismissed talk of being under pressure.
"I just do my tournament. Then the French players who play well and win matches, it's fantastic for the country, especially because it's in France," he said.
"So for the audience supporting us and cheering us, it's good to offer them victories. That's what they are expecting. And we are happy to offer them that.
"But then I'm just playing my tournament."
Under the guidance of his new coach Emmanuel Planque, Moutet trained with 20-times Grand Slam singles champion Federer in Dubai last December, having also spent time at Nadal's academy in Spain.
"I learned things there. I learn things every day where I am. I will continue learning, and I will still continue learning in the future. These were beautiful experiences," Moutet said.
"I thank those who welcomed me at the academy. Roger, who welcomed us in Dubai, was really a great help. So it's fantastic to have the support of these players, to have them helping me, and it was great to share, to observe.
"And these are the best players of all time so it's a fantastic experience."
(Reporting by Julien Pretot; Editing by Toby Davis)
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