Vondrousova, who became the first unseeded player to win Wimbledon last year, was a beaten finalist at Roland Garros in 2019, and the Czech player kept alive her dream of a return to the final with her straight sets win.
The fifth seed began strongly, breaking to love in the opening game, and although Paquet responded immediately with a break of her own, Vondrousova went on to win the next five games to take the first set with ease.
Paquet, 29, had never reached the third round before and her struggles continued in the second set when Vondrousova took a 3-0 lead, but the French player rallied and broke to pull it back to 4-3 down.
Vondrousova responded and broke again to move within one game of victory, and although Paquet forced a break point, Vondrousova held serve and moved into the fourth round.
The Czech will now play Serbian qualifier Olga Danilovic, who has already knocked out 11th seed Danielle Collins.
"Yeah, I saw the whole match because I was playing after them, so I saw everything," Vondrousova said.
"It was very tough, and she's a great player. I know her since we were juniors, so yeah, it's going to be very tough on clay.
"She's doing an amazing job, it's also fourth round and second week, so I think anything can happen now."
Vondrousova and Danilovic have met only once before, with the Czech player coming out on top when they faced each other at the United Cup in January.