The 37-year-old Monfils was coping well with the power play of the talented Spaniard until he hurt his ankle in the fifth game.
While Monfils carried on bravely, bringing out some wonderful defensive shots from Alcaraz, the world number two was too much for the Parisian.
There were plenty of smiles, in between the occasional wince from Monfils, who enjoyed some entertaining rallies where Alcaraz could show his deftness of touch.
With the outcome feeling like a formality, Alcaraz eased off the gas and Monfils was able to break for the first time to reduce the deficit to 5-3.
Monfils then held his serve with a brilliant cross-court winner before Alcaraz wrapped up business to book a place in the fourth round where he will face Lorenzo Musetti, who beat Ben Shelton.
"He's a great athlete," Alcaraz said of Monfils. "He reads almost every ball so I had to be patient but at the same time, with my forehand, my best shot, try to move him around the court, to get him tired a little bit and give myself the chance to dominate the point, get to the net, hit my best shot.
"That's what I was trying to do in the match and it worked pretty well," he added.
