The German sixth seed won 7-5, 3-6, 6-3, 4-6, 7-6 (10/3) in a match lasting more than four hours and will face either second seed Carlos Alcaraz or unseeded Serbian Miomir Kecmanovic in the last eight.
"At the end of the day it's a Grand Slam and everyone is playing at their best here," said Zverev. "Cam has definitely been playing amazing tennis and I'm just happy to get through."
Both men were rock-solid on serve in the first set until Zverev won a compelling 16-shot rally to earn two break points in the 11th game, making the vital breakthrough when Norrie netted.
The German, who had won all four of their previous Tour-level matches, broke first in the second set to ramp up the pressure on his 19th-seeded opponent but Norrie then went on a four-game winning spree to level the match.
A single break was enough to give Zverev the third set and the roles were reversed in the fourth set as Norrie hit back.
The feel-good factor rapidly evaporated for the British number one, who dumped a forehand into the net to lose his serve in the opening game of the deciding set.
But he was back on level terms within minutes, breaking for the fourth time in the match.
A wild forehand from Norrie in the seventh game handed Zverev two more break points but he failed to capitalise.
The deciding set almost inevitably went to a tie-break, which turned into a one-sided affair. Zverev raced into an 8-2 lead and he closed it out.
The German has a career-high ranking of number two but has never won a Grand Slam.
He was runner-up at the 2020 US Open and has reached the semi-finals at the Australian Open and the French Open.
