Rybakina was unable to defend her Italian Open title through illness and after losing serve in the opening game at Roland Garros and going 2-0 down, it looked like the Kazakh could be in trouble.
Rybakina withdrew from the French Open last year before her third-round match.
"Last year I missed this tournament and it was really hard but really happy to be back and compete here," Rybakina said.
"I'm really happy, thank you so much for coming guys, it's been an amazing atmosphere."
After Minnen's fast start, fourth seed Rybakina settled down to take the next six games and win the first set and the Kazakh began the second in the same fashion as she cruised into a 4-0 lead.
Minnen rallied and broke serve, and when Rybakina broke for a third time in the second set and served for the match, the Belgian broke again. But it only delayed the inevitable as the Kazakh held her next service game to win the match.
The roof was in use again on Court Suzanne Lenglen, as the rain fell on Roland Garros, but Rybakina was untroubled by the conditions.
"Of course, the ball is a little bit heavier and you need to move better and it's different, but that's tennis and you need to adjust with this rainy weather," she said.
Rybakina is the only player to beat Iga Swiatek on clay this year, defeating the world number one and three-times French Open winner on her way to the Stuttgart title in April.
She will now have a couple of days rest before facing either Arantxa Rus of the Netherlands or Germany's Angelique Kerber.
