Garcia, the 21st seed, shook off a sluggish start during which she dropped the opening set to eventually get past German Eva Lys 4-6, 7-5, 6-2 under the roof at Philippe Chatrier.
"It's never easy to start a new Roland Garros, there have been difficult moments for me in this tournament, even if it's dear to my heart," said Garcia, who is bidding to become the first French woman to win the title since Mary Pierce in 2000.
"I succeeded to stay in the game and to get better as the match goes on. It's difficult for me to keep my focus at a high level, especially with my game."
Up next for the 2022 WTA Finals champion, who has lost in the second round in her last three French Open appearances, is American Sofia Kenin.
At 37 and in the twilight of his career, Gasquet is not the man that most fans will be pinning their hopes on of ending the nation's wait for a first homegeown champion since Yannick Noah's triumph in 1983.
But in his 21st Roland Garros main draw appearance that equalled Feliciano Lopez's Open era record, the silky Gasquet showed plenty of steel to beat Croatian Borna Coric 7-6(5), 7-6(2), 6-4 to huge cheers from the Suzanne Lenglen crowd.
"I knew that the first set was key for me. I handled the decisive moments well, the first and second set tie-breaks," said an emotional Gasquet.
"It may not be the best moment of my career, but still a superb moment."
Humbert, the highest seeded French player in the men's draw was defeated 4-6, 6-2, 4-6, 3-6 by Italian Lorenzo Sonego, marking the fifth time in six appearances that he fell at the first hurdle in the year's second Grand Slam.
The Marseille and Dubai champion has struggled in the clay swing with his form steadily dipping and he was exposed again by Sonego in a rematch of their second round meeting last year.
"He's really good on clay, that's true. His match was really very solid. My level wasn't as good as I expected and therefore, I thought, well, it's not enough to go to the second round," Humbert told reporters.
Backed by a raucous crowd on Simonne Mathieu court, Corentin Moutet overcame Rome runner-up and 16th seed Nicolas Jarry 6-2, 6-1, 3-6, 6-0.
Chloe Paquet upset 6-3, 6-1 Russian Diana Shnaider to reach the second round, seven years after her first and only win in the French Open main draw. She will meet the Czech Republic's 32nd seed Katerina Siniakova.
Alexandre Muller defeated Italian Luca Nardi 6-4, 6-1, 6-1, for his first win in his fourth Roland Garros appearance.
