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.”
Draper bows out
Jack Draper was forced to retire in the second set of his fourth-round clash with Carlos Alcaraz.
The 21-year-old Briton was ultimately overcome by an abdominal injury he picked up during his previous match against Andy Murray.
The result means Alcaraz moves within three wins of regaining the world number one ranking.
“I was doing further damage, so there was no point in making it worse,” the 21-year-old told BBC Sport. “At the end of my match against Andy [Murray, on Monday], I obviously had a bit of hip pain but my ab towards the end was getting sorer.
“I woke up [on Tuesday] and my ab was sore and when I started serving 100 per cent in the match it gradually got worse. I couldn’t really play properly. I’m at the start of properly working hard on my fitness training.
“I’ve done a lot of stuff on court over the years but I’ve never really worked truly on my body.
“I’m still early in my development. I’m starting to play grand slams with five sets. The physical nature of this sport is very demanding and so you’ve got to have a really resilient body to cope with the demands of the tour.”
Raducanu suffers heavy defeat
World number one Iga Swiatek swept past Emma Raducanu in straight sets to bring the Briton’s valiant BNP Paribas Open run to an end in the fourth round.
Raducanu, who has battled a wrist injury and illness in recent weeks, showed glimpses of her US Open-winning best form throughout her earlier matches, with victories against 20th seed Magda Linette and 13th seed Beatriz Haddad Maia particular standouts.
The 20-year-old looked impressive again in the early stages against Swiatek, matching the Pole from the baseline as neither player looked close to giving up a break.
However, the turning point came in the sixth game of the opening set.
Down 3-2, Raducanu double-faulted at 0-30, giving defending champion Swiatek the space she needed to claim the decisive break.
The world number 77 was simply outmatched from there, ultimately going down 6-3 6-1 in an hour and 24 minutes.
“I think overall I can say it was a positive week for me,” said Raducanu. “The team, I think we had a run of three really good matches.
“It was a good experience today playing Iga on that court at night. I think the conditions are very slow and heavy and favoured her. But, yeah, it’s just good to see where I’m at.
“I think it’s been a good 10 days. I haven’t trained particularly much before the tournament, but I feel like I’ll probably take a day to recover tomorrow physically, because since the tournament started, I have done a little bit of something every day.
“So (I will) reset a bit mentally and physically tomorrow, and then take it from there.”
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