Norrie, who has now claimed 44 wins this year, will face Andrey Rublev in Monday's fourth round, after the Russian outclassed Denis Shapovalov 6-4, 2-6, 7-6 (3), 4-6, 7-6 (7).
The British number one, who is yet to drop a set, felt his 7-5 6-4 6-1 victory over young Dane Rune was his best performance so far.
Norrie, who next plays ninth seed Andrey Rublev, said: "I thought I played a bit better than I did in the previous match, so that was great. I felt way more comfortable. It was nice to be on Grandstand.
"It was tough playing against Holger. He can come out and play very aggressive and can come forward very well.
"He's extremely talented, and he's a young up-and-coming player, but it was nice to get through in straight.
"I think it's great with the seeding. You can work your way more into the tournament than going out there, having to play a top seed early on. I've still got a lot of things I would like to improve, and I've got tomorrow to do that."