The fifth seed caught the Italian at a down moment, with Sinner appearing low on energy in the final set and frequently flexing his right hip.
It was the second match of the day for both Sinner and Rublev as organisers made up for a rained-off Friday.
The Italian top seed played this week for the first time since Wimbledon, where he fell ill and exited in the quarter-finals before missing the Paris Olympics with tonsillitis.
Rublev credited a quick snack from his trainer for reviving him during a 30-minute rain interruption after winning the opening set.
"Maybe my sugar levels were low. But my trainer gave me a sugar bar, I ate it and one game later I was perfect again," Rublev said.
"I played really well, I'm happy to win. Jannik is an amazing player. He has been playing unbelievable the past two years.
"I was hoping to fight him and show some great tennis."
Sinner's form took a dive in the deciding set, which he lost on a double-fault after saving two Rublev match points.
Rublev owns two defeats of number one players against four defeats. He also beat top-ranked Novak Djokovic at Belgrade two years ago.
Sinner started his day with a 6-4, 6-3 third-round victory over Canadian-born Chilean Alejandro Tabilo, advancing in 80 minutes at the tournament, which ends on Monday due to Paris Olympic scheduling.
Rublev had earlier defeated American Brandon Nakashima 6-2, 6-2 in a third-round match.
Second seed Alexander Zverev, the 2019 champion in Canada, overcame stiff late resistance from Dane Holger Rune, winning a third-round match 6-3, 7-6(5) with 21 winners including eight aces.
"With this wind, you won't be playing pretty tennis," Zverev said. "I had to find a way to win. I did that today - and we move on."
Zverev leads the ATP with 49 match wins this season.
Sinner, who stands 24-2 this season on hardcourt, has qualified as first into the field at November's ATP Finals in Turin.
Former US Open finalist Kei Nishikori extended his comeback story, saving eight of 10 break points in a 6-3, 6-4 defeat of Portugal's Nuno Borges, last month's Bastad champion.
But Nishikori missed out on his first ATP semi-final since 2021 in Washington when he dropped a 6-4, 7-5 quarter-final to Italy's Matteo Arnaldi.
It was the second win of the day for Arnaldi, who earlier advanced 4-6, 7-6(5), 3-0 when Spaniard Alejandro Davidovich Fokina retired with stomach illness.
Hurkacz wins twice
Delayed second-round matches were wrapped in early afternoon with recovering Hubert Hurkacz leading the way with his 4-6, 6-3, 7-6(6) fightback win over qualifier Thanasi Kokkinakis, the fourth seed's first outing since knee surgery.
Poland's Hurkacz had not played since quitting injured in the Wimbledon second round and then heading home for a knee operation, with doctors telling him his season was over.
"I'm happy I'm able to compete and I wasn't that slow today," he said. "Since this was the first match since the surgery, it was a little bit unknown. "I'm feeling quite decent. I spent a lot of last week in the gym."
The Pole had double duty also, coming back in the night session to defeat France's Arthur Rinderknech 4-6, 6-3, 6-4.
Last week's Washington title holder Sebastian Korda was a 6-4, 7-6(4) winner over fellow American Taylor Fritz and then reached the quarters by walkover when sixth seed Casper Ruud was forced out by illness.
Korda will face Zverev next.
