Draper was heavy favourite to reach the final eight at the Foro Italico for the first time after a scintillating season that saw him clinch his maiden title at Indian Wells and finish as runner-up at the recent Madrid Open.
But after losing the opening set, Britain's number one stormed back to overcome the Frenchman in two hours and 19 minutes at a sun-drenched SuperTennis Arena.
The 23-year-old angily smashed his racket on the court in his last match against Vit Kopriva and appeared agitated once again by his lackluster start in front of a partisan crowd.
Draper was "bamboozled" by Moutet's drop shots and brilliant defence, forcing him to make 15 unforced errors as he huffed and puffed his way around the court.
"I felt very uncomfortable. I didn't really know what to do," he said. "I don't think I've played many people like him before. Credit to him. I was finding it so strange from the start."
Yet Draper regained his composure and dug deep to win the next two sets and book his passage into the last eight.
"I knew I needed to change a few things tactically, had to be there for every point because his level wasn't going to drop," said Draper.
"Really, really happy with the way I problem solved. And I think there's a lot of lessons I can take from that.
"He frustrates players massively because there's not many like him. His drop shots are incredible. He was moving around his backhand really well. I was caught off guard by what he was bringing."
Moutet overcame Holger Rune in a grueling match lasting almost four hours on Sunday but there were no signs of fatigue against his top-five opponent as he took a pulsating first set in 39 minutes.
However after losing a video review for a possible foul in the following set, the match slowly ebbed away from Moutet and the result was never really in doubt after he received medical treatment to his left hamstring in deciding set.
"I'm disappointed because I felt like I had the match and I could win it," Moutet said.
"He deserved to win it, but I felt like I could do a bit better, stay more aggressive until the last point. At this level, if you’re not perfect in some details it's tough to win. I have some regrets on some shots, but congrats to him.
"He's very aggressive. He served good. His forehand is very powerful. You always have to put pressure on him because if I play an easy ball with a bit less intensity, I'm going to run so much.
"It was the first time I play with him. He's very, very intense."
Draper and Alcaraz set to clash on clay
Draper will climb to number four in the world rankings if he reaches the semi-finals in Rome, overtaking Taylor Fritz who crashed out to Marcus Giron in the round of 64.
But he will face a stern challenge to his clay credentials from Carlos Alcaraz, who he beat in the semi-final at Indian Wells, after the Spaniard downed Karen Khachanov 3-6, 6-3, 7-5 in their last-16 match.
"I suppose we're both in the top five in the world now," Draper said of their new rivalry. "I see it as just two guys who are playing great tennis, going out there and wanting to win and beat each other and be competitive.
"I have big respect for him. I know what he's going to bring. He's going to be really tough to beat."
Alcaraz leads their head-to-head record 3-2 but is wary of the threat an in-form Draper poses, especially after his recent defeat in California.
"He's really solid," Alcarez said. "In every point, his shots are great, really aggressive.
"He's trying to play as aggressive as he can and that makes him really, really tough. Every point is a battle. Physically he improved a lot; with the serve, with everything. He's a really, really complete player.
"I'm just looking forward to playing him. He's one of the best players in the world right now. The results show how good he is playing lately. I just want to take revenge for Indian Wells."