The 19th seed emerged from an excruciatingly tight deciding set to battle through 2-6, 6-3, 7-6 and will face fellow American Sebastian Korda in the quarter-finals.
Shelton, who lost to Novak Djokovic in the last four in New York last month, did his now-trademark "dialled-in" celebration to mark his first time into the quarter-finals of a Masters.
The 26th-seeded Korda fended off a late fightback from the 20th seed Francisco Cerundolo of Argentina to go through 7-5, 7-6.
Shelton, who turned 21 on Monday, endured a torrid opening set against the in-form sixth seed Sinner, summed up by his double fault to concede the seventh game of an error-strewn first stanza.
The Italian Sinner, fresh from winning the China Open in Beijing, clinched the opening set with a well-placed smash.
The athletic Shelton was a changed man in the second, sealing the set with an ace and a roar, and then immediately put Sinner under the cosh to start the decider.
The 22-year-old Sinner survived the initial onslaught, saving five break points to seal a game that lasted nearly 10 minutes, and the set went with serve to force a tiebreak.
But it was Shelton who prevailed - and a crowd that had been on his side serenaded him by singing Happy Birthday.
Earlier in the day, Fabian Marozsan pulled off a shock win against Casper Ruud.