The world number eight won 6-3 1-6 6-2 against the Italian top seed in a rain-interrupted match in Montreal.
Sinner struggled with an injury to his right hip and the 22-year-old frequently massaged the area, which could be a concern to him with the US Open starting in a fortnight’s time.
“It was a really great match of mine. Jannik is an amazing player and he has played unbelievably for two years. I was just hoping I would be able to fight with him,” the 26-year-old said.
Rain wiped out Friday’s play so both men had played their last-16 matches earlier in the day, with Sinner defeating Chile’s Alejandro Tabilo 6-4 6-3 and Rublev seeing off American qualifier Brandon Nakashima 6-2 6-2.
Rublev will face another Italian, Matteo Arnaldi, in the semi-finals after he beat Japan’s Kei Nishikori 6-4 7-5.
In the women’s tournament in Toronto, defending champion Jessica Pegula beat fellow American Peyton Stearns 6-4 7-5 to reach the semi-finals.
Pegula will play Diana Shnaider in the last four after she won 4-6 6-1 6-4 against fellow Russian Liudmila Samsonova.
Amanda Anisimova of the United States – who returned to tennis at the end of 2023 following a ninth-month break because of mental health issues – upset Belarus’ Aryna Sabalenka 6-4 6-2.
Anisimova will face Emma Navarro, who defeated another American Taylor Townsend 6-3 7-6 (7-5).
The US Open in New York starts on 26 August.