The tournament's top seed, world number five Novak Djokovic, also reached the last four with a 7-6(1), 6-4 win over Nuno Borges.
Italian Musetti, 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 qualifies will find himself in a group with world number one Jannik Sinner, as well as Alexander Zverev and Ben Shelton.
Musetti will face the American Sebastian Korda in Friday's semi-finals.
Djokovic claimed his 200th indoor win after one hour and 44 minutes and will play German qualifier Yannick Hanfmann for a place in the final.
The 38-year-old Serb has already qualified for Turin and has been matched with Carlos Alcaraz, Taylor Fritz and Alex de Minaur.
He is bidding for his 101st career title in Athens. Jimmy Connors leads the way on 109, with Roger Federer second on 103.