The world number nine will face either Arthur Fils or Brandon Nakashima after a 6-3, 6-4 win.
He is the highest remaining seed left in Barcelona after Carlos Alcaraz pulled out of the tournament on Wednesday with a wrist injury.
Musetti impressed on clay last season, reaching at least the semi-finals in the ATP 1000 events in Monte Carlo, Madrid and Rome, as well as at the French Open.
But he has struggled for form and fitness this year and is still without an ATP title since 2022.
Musetti had lost in his opening matches at both Indian Wells and the Monte Carlo Masters, after being forced to retire injured from his Australian Open quarter-final against Novak Djokovic in January.
But after beating Martin Landaluce for his first win since Melbourne on Tuesday, he was in fine fettle against Moutet and much too strong for his 31st-ranked opponent.
Earlier Thursday, Russian fifth seed Andrey Rublev saw off Lorenzo Sonego 6-2, 6-3 to set up a quarter-final meeting with Tomas Machac, the beneficiary of Alcaraz's withdrawal.