The Italian 25th seed, who showed his prowess on grass by reaching the final last month at Queen's, broke the French lucky loser five times on Court No. 2.
Mpetshi Perricard, Wimbledon's ace leader with 115 aces, was rock-solid on serve in the first set, taking it with a single break, but stumbled towards the end of the second set, allowing Musetti to level.
An early break for the Italian in the third set put him in the driving seat and the Frenchman ran out of steam.
Musetti hit a total of 23 winners against just eight unforced errors, compared to his opponent's tally of 42 unforced errors.
"It's tough for me to get emotional but I think today I will," he said. "I've been dreaming of this moment since I was a kid.
"It's such a big day for me and I'm really happy and proud of this win against a tough opponent. I struggled a little at the beginning against a big serve. I'm a little bit shaking still but at the end it's a really phenomenal day for me."
Musetti will face either German fourth seed Alexander Zverev or Taylor Fritz of the United States for a place in the semi-finals.