The eight-time Wimbledon champion had a slight wobble in the third set when he missed two break points at 5-5 and was then broken at love by Norrie.
After trading breaks in the fourth set, Federer broke again for a 5-4 lead.
He will play No. 23-seeded Lorenzo Sonego of Italy on Monday.
Norrie was the last British man remaining in the tournament.
Nick Kyrgios had to retire with an abdominal injury after the second set of his third-round match against No. 16 Felix Auger-Aliassime. Kyrgios told a trainer he couldn't serve properly and was worried about tearing a muscle.
"It's getting worse and worse," Kyrgios said. "I'm trying but I can't even ... I can't serve anymore. I can't do it."
Kyrgios won the first set 6-2 but lost the second 6-1 on No. 1 Court.
The Australian is also playing mixed doubles with Venus Williams.
The men's singles started on a more light-hearted note when Kyrgios stepped onto the court only to realize he left his grass-court shoes in the locker room.
His mishap forced the two players to stand around for a couple of minutes before they could start their warm-up.
"The one day I thought I was being a professional," Kyrgios joked.
A Wimbledon employee reunited him with the right shoes.