World number 69 Moutet, who benefited from the withdrawal of Germany's Jan-Lennard Struff due to injury on Tuesday, could not cope with the American's strong display of serving and precision at the net on the red clay of Roland Garros.
Paul broke Moutet's serve twice in the opening set but the Frenchman managed to fight back each time, saving a set point at 5-4 with a breath-taking passing shot. However, world number 13 Paul stepped up his game to claim the tie-break.
The American quickly opened up a lead in the second as Moutet chained together some unforced errors. The Frenchman battled gamely but after saving six match points he sent the ball long, putting Paul into the last eight.
"I really believed in myself until the last point. It's never over. I was pretty close to breaking in the last game, so I really believed in it," said Moutet, who dropped out of this month's Wimbledon due to injury.
"I knew that if I lost today, we wouldn't have any chance to take a medal but it was a big motivation as well," he said.
Paul made his Olympic debut at Tokyo 2020 where he fell in the first round of the singles.
The American could meet Carlos Alcaraz in the quarter-finals after losing to the Spaniard at the same stage of Wimbledon this year. Alcaraz faces Russian Roman Safiullin, the world number 66, who is competing as a neutral athlete, later on Wednesday.
