Three-time Roland Garros champion Djokovic saved a set point on serve in the third set to seal a straight-sets victory over home favourite Moutet, prevailing 6-3, 6-2, 7-6(1).
It was a surprisingly flat and uneventful match for the most part, with Moutet unable to consistently challenge the world No. 6, the world No. 73 falling to a 0-3 deficit in their head-to-head.
"I had to really concentrate in a match like that," said Djokovic in his on-court interview. "It was a real battle, particularly in the third set.
"Then he had set point in the 12th game, I knew that anything was possible, but I knew I had the energy to stay focused and finish it.
"I tried to stay calm in the pressure moments, but it wasn't easy."
Djokovic appeared in cruise control throughout the majority of the match on Court Suzanne Lenglen, moving two sets to love up with relative ease.
However, issues then began to emerge for the 38-year-old in the third set.
Djokovic took a medical timeout due to what appeared to be blisters on his foot, and was then broken by Moutet when action resumed.
The Serbian appeared to intentionally attempt to keep the rallies as short as possible to ease any pain, though he was able to break straight back and ultimately close out the match.
Victory for Djokovic was his 98th at Roland Garros, and he is now just two victories away from sealing a staggering 100 match victories at the tournament.
Only Rafael Nadal (112) has hit that milestone previously in Paris, and, should Djokovic hit that feat, this would become the first Grand Slam tournament he has recorded 100 match wins at.
He came close to that milestone back at the Australian Open in January, recording his 99th match win with a victory over Carlos Alcaraz in the quarter-final, before retiring injured versus Alexander Zverev.
Djokovic has now also reached at least the third round of the French Open for the past 20 editions of the event, a second-round defeat on his 2005 debut the earliest Roland Garros exit of his career.
He is the first man to reach the third round of the French Open 20+ times and only the second man across all four Grand Slams, following Roger Federer (21) at the Australian Open.
The Serbian will next face Filip Misolic, who upset 27th seed Denis Shapovalov in a five-set thriller.