The Russian never got going in the second-round tie as Popyrin eased to a 6-4, 6-4 win at the traditional clay court warm-up for the French Open.
The world No 46 will face compatriot Alex de Minaur, who beat Tallon Griekspoor in three sets, for a possible quarterfinal against Novak Djokovic.
“I am feeling really comfortable on it and happy to beat a guy who was in form, confident and the defending champ. It was an awesome match.”
The loss extended a poor run of form for Rublev, after a defeat in his opening match against Tomas Machac in Miami and losing to Jiri Lehecka in the Indian Wells last 32.
The sixth seed had a break point to level at 5-5 in the second set, but found the net and Popyrin closed out the victory.
Rublev lifted his first Masters 1000 title in Monaco last year by beating Holger Rune in the final.
Australian Open champion Sinner cruised through, though, seeing off American Sebastian Korda 6-1, 6-2.
The world No 2 converted seven of the 12 break points he created and saved all three of the ones he faced in a dominant performance.
The Italian will next play Jan-Lennard Struff of Germany in the last 16.
“Every year, it’s tough to come here and try to perform well, but I’m very happy about this performance and lets see what is coming in the next round,” said Sinner, who lost to Rune in the semifinals 12 months ago.
“I haven’t served very well, but I returned very well, I stayed close to the baseline.”
Two-time champion Stefanos Tsitsipas showed signs of a return to form, thrashing Argentina’s Tomas Martin Etcheverry 6-1, 6-0 in just over an hour.
French 14th seed Ugo Humbert progressed with a 6-1, 6-4 success against China’s Zhang Zhizhen.
He will next take on Lorenzo Sonego, who ensured former top-10 player Felix Auger-Aliassime’s miserable run of form continued, winning 6-4, 7-5.
Italy’s Sonego only reached the main draw as a lucky loser after the withdrawal of Carlos Alcaraz.