The British No1 needed just 75 minutes to see off the Russian sixth seed 6-2 6-4 on Wednesday night 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 No2 Carlos Alcaraz to secure Rio Open glory last month, will face either American Frances Tiafoe or Chile's Alejandro Tabilo 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.
Additional reporting by the Press Association.