The Spaniard, who turns 36 next week, will next meet Dutch 26th seed Botic van de Zandschulp with a potential quarter-final looming against world number one Novak Djokovic who defeated him in last year's semi-finals.
Playing under the lights at Court Philippe Chatrier, Nadal peppered the red clay with winners from both his forehand and backhand in the first two sets to leave Moutet stranded.
In his first career meeting with the 23-year-old French wildcard, Nadal broke Moutet's serve four times in the first two sets but uncharacteristic errors at the start of the third allowed his opponent to take a 2-0 lead.
Nadal immediately broke back, however, to get the set back on serve and after another exchange of breaks, the Spaniard broke Moutet again to seal victory on his first match point.
Grand slam win No. 300 for Rafael Nadal.
— Roland-Garros (@rolandgarros) May 25, 2022
"Preparation hasn't been perfect so I didn't expect a perfect start to the tournament," Nadal said.
"But it has been going quite well. I am positive about the things that I did out there tonight. Of course, there is room to improve and I need to improve if I want to keep having chances to go deeper."