Sinner, despite a less-than-sparkling performance, recorded his 29th consecutive indoor hard court win and secured top spot in his group, while Shelton is out after losing all three of his group matches.
The Italian will now face Alex de Minaur in the semi-finals, an opponent he has defeated in all 12 of their previous matches.
In the men's doubles, Briton Henry Patten and his Finnish partner Harri Heliovaara progressed to the semi-finals with a 7-6(5), 6-2 win against Marcelo Arevalo and Mate Pavic.
Patten and Heliovaara will play home favourites Simone Bolelli and Andrea Vavassori, with an all-British final still possible as Lloyd Glasspool and Julian Cash face Neal Skupski and Joe Salisbury in the other semi-final.