The Italian 25th seed struggled against the big-serving Frenchman in the first set before adapting his approach to win 4-6 6-3 6-3 6-2.
It ends a remarkable run for 21-year-old Mpetshi Perricard, who was attempting to become the first lucky loser, man or woman, to reach the singles quarter-finals of a Grand Slam.
“It is a really phenomenal day for me,” an emotional Musetti said in his on-court interview.
He will next face either French Open finalist and fourth seed Alexander Zverev of Germany or American 13th seed Taylor Fritz.
Playing on his 21st birthday, Mpetshi Perricard entertained the court two crowd in the early stages with trick shots and huge serves.
He lost in the final round of qualifying before being handed a reprieve after withdrawals from the main draw, and since then had made eye-catching progress.
Musetti seals victory after longest rally thus far
Standing 6ft 8in and using that height to his advantage, Mpetshi Perricard’s positive approach brought the reward of a break at 3-3 and he calmly cruised through the remainder of the opening set.
But Mpetshi Perricard’s confidence began to falter as Musetti settled and got to grips with his opponent’s first serve, clinically moving through the next three sets and making just eight unforced errors in the match.
The 22-year-old fell flat on his back after wrapping up the landmark victory with his third match point, before holding back tears in his on-court interview.
“It is tough for me to get emotional, but today I think I will,” Musetti said.
“I’ve been dreaming of this moment since I was a kid. I always had a really beautiful family that always supported me, choosing my dream.
“It is tough to speak.”
The win continues an impressive grass-court swing for Musetti, who was a runner-up to Tommy Paul at Queen’s Club last month.