The world number one easily won his 25th straight match, taking just over an hour to see off Australia's Popyrin and set up an Italian derby with Andrea Pellegrino in the next round.
Pellegrino, 29, who beat 20th seed Frances Tiafoe 7-6(8), 6-1, is a qualifier ranked 155 in the world who, before this week, had never played in the main draw of a Masters 100 tournament.
Sinner, meanwhile, is trying to extend his own record-breaking run of overall victories in the ATP's top-ranked events after having won his last five.
The 24-year-old joins Novak Djokovic as the only player to win a year's first 25 Master 1000 matches, with the Serbian tennis icon's record run stretching to 31 matches in 2011.
Popyrin was suffocated by Sinner and made 23 unforced errors on his way to a comprehensive defeat.
The world number 60 only got 48% of first serves into play, and that allowed Sinner to break serve five times and close out a match which was barely a contest.
"It has been a very good performance from my side," said Sinner.
"He's a big server so his percentage was not very high, which helped me for sure a little bit, but I've been returning very well the second serves."
Should Sinner prevail at the Foro Italico, he will be the first Italian to win there since Adriano Panatta 50 years ago, and would complete his collection of Master 100 tournaments.
With great rival Carlos Alcaraz out injured and Novak Djokovic eliminated early, Sinner will be red-hot favourite as he builds towards completing the career Grand Slam at the French Open which starts next week.
Former Rome champion Daniil Medvedev gets his tournament underway against Pablo Llamas Ruiz after a second-round walkover gave the Russian seventh seed automatic passage into the third.
Gauff survives scare
Coco Gauff survived a scare against Iva Jovic, coming back from a match point to win 5-7, 7-5, 6-2 and reach the women's quarter-finals.
American star Gauff, who is the reigning French Open champion and reached the final in Rome last year, won a draining match which took the best part of three hours.
The 22-year-old has struggled on clay this season, with a last-16 exit in the Madrid Open coming after a disappointing run to the quarter-finals in Stuttgart.
Gauff was so annoyed with herself shortly after saving a match point against the serve at 5-4 down in the second set that she smacked herself in the head.
But she recovered to eventually see off her fellow American and set up a clash with Mirra Andreeva in the last eight.
"It was really hard," said Gauff.
"I think the conditions made it tough to make some clean tennis. Really proud of how I was able to fight."
Later, Naomi Osaka will face three-time Rome champion Iga Swiatek, who has not won a clay tournament since capturing the last of her four French Open titles in 2024.
Reaching the last eight would be the best result of a difficult season for Osaka, a four-time Grand Slam winner, after being knocked out of Indian Wells and the Madrid Open in the last 16 by Aryna Sabalenka.