The Italian, ranked ninth in the world, needs to win the tournament to overhaul Felix Auger-Aliassime and claim the final place in next week's showpiece in Turin.
Whichever one qualifies will find himself in a group with world number one Jannik Sinner, as well as Alexander Zverev and Ben Shelton.
Musetti will now face the American Sebastian Korda in Friday's semi-finals.
The tournament's top seed, world number five Novak Djokovic, is also looking to reach the last four as he takes on Nuno Borges later on Thursday.
The 38-year-old has already qualified for Turin and has been matched with Carlos Alcaraz, Taylor Fritz and Alex de Minaur.