The ninth-seeded Russian triumphed 6-4, 6-4, 6-4 and will face either four-time champion Rafael Nadal or Frances Tiafoe of the United States for a place in the semi-finals.
Rublev lost to Nadal in the quarter-finals in New York in 2017.
"It was two and a half hours, it was tough," said 24-year-old Rublev who had needed four hours and five sets to get past Denis Shapovalov in the third round.
"Cameron and I, we have known each other for a long time. I knew I had to give my best.
"I played a good match. I was able to win in three sets so I am super happy,"
Rublev, who also made the last-eight in New York in 2017 and 2020, will be playing in his sixth quarter-final at the majors but has yet to progress further.
Monday's fourth round clash was delayed by half an hour after the roof on Louis Armstrong Court was closed too late to prevent a sudden downpour from soaking the surface.
The delay, however, barely threw Rublev off his stride as he sealed victory on the back of 11 aces and 30 winners.
Norrie, the seventh seed, was undone by 35 unforced errors.