It has been a superb season so far for Draper, claiming a maiden Masters 1000 title at Indian Wells, improving his fitness and finding another level of power behind his shots.
But he was the weaker player in the first set, capable of out-hitting Moutet but too often dragged out of position.
He hit 15 unforced errors to four from Moutet, who sealed the opening set with an emphatic ace.
Draper jogged to his chair at the end of the set to try and find some energy, often shouting “Jack, come on!” after an error and berating himself.
He steadied himself well after the video review, requested by Moutet at 4-3 in the second set. The video showed Draper had got his racquet head to the ball, with the frame causing the ball to bobble awkwardly and draw the error out of Moutet’s forehand in response.
Moutet briefly argued with the umpire and, perhaps distracted, played a poor next service game to be broken and allow Draper to serve for the set.
The Frenchman looked to have recovered when he broke Draper back to love, but Draper kept up the pressure and forced a deciding set on a long lob, with the Briton cupping his hand to his ear, urging more support.
Both served well in the third set but fatigue again played a part, with Moutet forced to take a medical timeout at 4-3 down after struggling with his thigh.
Draper promptly broke in the next game and then served out the match, closing it out with a well-timed serve and volley.
“I had a word with myself after the first set – I needed to really knuckle down and focus and I knew his level wasn’t really going to drop,” Draper added.
“I came out in the second set and fought for every point and found a way in the end to pick up my level.”
Earlier, British pair Joe Salisbury and Neal Skupski advanced in the doubles, beating India’s Rohan Bopanna and Czech Adam Pavlasek 6-3 6-3 to reach the semi-finals.
Monte Carlo champion Alcaraz advanced with a 6-3 3-6 7-5 win over Russia’s Karen Khachanov.
Meanwhile, world number one Jannik Sinner maintained his winning return from a doping ban with a 7-6 (7-2) 6-3 victory over Francisco Cerundolo.
The Italian has won all three matches since returning from a three-month suspension, but showed signs of rustiness in the opening set after exchanging breaks of serve with Argentine Cerundolo before closing out a tie-break.
The second set was much more comfortable and Sinner will face either Norway’s Casper Ruud or Spain’s Jaume Munar in the quarter-finals.
Adriano Panatta was the last Italian to win the men’s Rome title in 1976 and Sinner will be joined in the last eight by compatriot Lorenzo Musetti as they both bid to end the 49-year wait for a home champion.
Musetti engineered his way to match point against Daniil Medvedev before having to wait three hours to complete his victory as heavy rain forced the players from the court.
