Young Frenchman Fils, ranked 30th, will take on in-form Spaniard Rafael Jodar on Saturday after he downed Briton Cameron Norrie 6-3, 6-2.
Clay expert Musetti survived two break points in the eighth game of the first set, but the 21-year-old Fils took his third to claim a 5-3 lead and then served out.
Fils, who came back from an eight-month injury absence in February, has been in strong form, finishing as runner-up to Carlos Alcaraz in Doha.
He wrapped up the second set of the quarter-final clash with a break in the ninth game and then finished by winning to love on his serve.
"I was happy with the performance today, pretty solid throughout," said Fils.
"With Lorenzo, it is always very tough because he can change the rhythm."
Jodar, 19, has won all three of his matches this week in straight sets, including against world number 24 Norrie.
Ranked 55th, Jodar broke three times in a topsy-turvy first set, as well as conceding two breaks of his own.
The Spaniard picked off Norrie in the second set with breaks in the fifth and seventh games to triumph.
"We want more, we have to continue, it will be a new battle tomorrow and thanks to the fans for coming to support me," said Jodar.
"I'm a calm person and that mentality helps me handle games like this."
Earlier, fifth seed Andrey Rublev eased to a 6-4, 6-3 win over Tomas Machac, and he will face qualifier Hamad Medjedovic, who ousted Alex de Minaur earlier in the week, after he defeated Nuno Borges 7-6(6), 6-2.
World number two Alcaraz withdrew with a wrist injury on Wednesday and on Friday pulled out of next week's Madrid Masters.
