The British No.1 produced an impressive 6-4 6-7 6-4 6-3 win over 11th-seed Casper Ruud and will next face world No.6 Alexander Zverev or Alex Michelsen.
Norrie is the last British star left standing following the exits of Andy Murray, Jack Draper, Emma Raducanu, Katie Boulter and Jodie Burrage.
The 28-year-old, seeded 19th at Melbourne Park, produced his best Grand Slam run at Wimbledon two years ago with a run to the semi-finals.
Speaking in his on-court interview after victory over Ruud, Norrie was quizzed about the British support at John Cain Arena.
‘There was all sorts of nicknames being shouted out there, ‘Come on Nozzdog’ and then ‘Nozz’ and all things like that,’ Norrie said.
‘It was good stuff and I think everyone was enjoying themselves. I definitely could tell you guys had a few drinks on this Saturday afternoon! It was really good, a good atmosphere.’
Asked about reaching the fourth round of the Australian Open for the first time, Norrie added: ‘Honestly so special, especially on this court. I didn’t realise it was going to be this atmosphere and you guys were behind me from the very beginning.
‘I had to play really brave and almost won that second set. Casper served well, fought hard and I didn’t let it really bother me. It was a good mental performance to come out with a lot of energy in the third set.
‘I just kept a really good level throughout the match so I really enjoyed it. I stuck to my game plan and I got through.’