The Italian won his last 16 clash with the Swiss veteran 4-6, 7-6 (7/5), 6-4 to earn the second seed a last eight match-up with Alexandre Muller.
Musetti needs a run to the final in Greece to deprive Felix Auger-Aliassime of the last ticket to Turin.
The Canadian moved into the eighth and final spot for the end-of-season event starting Sunday as a losing finalist at last week's Paris Masters.
Athens top seed Novak Djokovic returned after three weeks out of action to ease past Alejandro Tabilo in straight sets on Tuesday and progress to the quarters.