A day after overcoming world No 13 Matteo Berrettini, Murray prevailed 7-6(6) 6-7(4) 6-1 in his first meeting with the British No 3.
Murray’s victory in two hours and 53 minutes sets up a last-eight showdown with Dominic Thiem.
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After winning two tie-breaks to beat Berrettini on Monday, Murray admitted playing matches on back-to-back days would truly test his fitness as he continues he comeback trail.
And so it proved against Norrie, with the first two sets alone taking almost two-and-a-half hours.
The first set saw Norrie break back when Murray was serving for it at 5-3, but it was the former world No 1 who came out on top in the tie-break.
After trading breaks midway through the second set, Murray faced his fourth-straight breaker, but this time lost as Norrie forced a decider.
And though he seemed to be flagging, Murray had enough energy in the tank to breeze through the final set, breaking Norrie twice and dropping just three points on serve to advance.
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"I'm really tired, I just had a sleep before coming," Murray admitted afterwards.
"That's the first time I've had to do that since I came back. It's something that I need to kind of get used to again, especially playing at this level and that intensity.
"It was a good step for me, it's great that I won the match. Whether I'd won or lost it, I was able to come out... and be competitive and play some good tennis... I'm happy I've got the day off tomorrow to recover."
Murray's next opponent Thiem booked his place in the last eight with a 6-3 6-3 win over Zhizhen Zhang.
The Austrian fired five aces, won 84% of his first service points and needed only three breaks to beat Zhang in 69 minutes.
Italy's Fabio Fognini beat Russian Andrey Rublev 6-3 6-4 to set up a quarter-final clash with fourth seed Karen Khachanov, who dispatched Frenchman Jeremy Chardy 7-6(0) 7-6(5).
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