The 27-year-old needed just 75 minutes to see off the Russian sixth seed 6-2 6-4 and claim an eighth successive win with one of his best performances of the season to date.
Norrie, who won the tournament in 2021 and beat world number two Carlos Alcaraz to secure Rio Open glory last month, will face American talent Frances Tiafoe in the last eight.
In a match which started with three successive service breaks, it was the Briton who took the initiative to race into a 3-1 lead, breaking for a third time to give himself the chance to close out the first set.
Indeed, the world number 12 converted all five of his break points as he completed a victory which squared his ATP Head2Head series against 25-year-old Rublev at 2-2.
Speaking after the match, Norrie said: "Off the ground I was rock solid."
"It was a little cooler today so I felt like I had more time on the ball. I played well in the big moments. Andrey is not an easy guy to beat, so I was really pleased to come through in straight sets.
"There wasn't a lot to take from [my loss to him at the US Open last year]. I played one of my worst matches of the year. Maybe that's why he got a little frustrated today because what he did last time wasn't working today."
Meanwhile, Tiafoe defeated Chilean qualifier Alejandro Tabilo 6-4 6-4, marking his 150th tour-level victory.
"That 3-0 hold was huge," Tiafoe quipped afterward.
"That loosened me up. I played a great return game after that and I just took my level up to a whole other level and he wasn't able to match. At the end it got pretty breezy there, so it was tough to close it out, but other than that, after that 3-0 game I thought it was one-way traffic."