Anderson will face another Englishman, fellow former champion Greg Harlow, who edged out Scotland's David Gourlay.
Scottish four-time champion Paul Foster faces England's Les Gillett in Saturday's other semi-final.
A delighted Anderson told BBC Sport: "Mr Paul Foster has just told me he has never seen me play better."
The 2013 champion swept the world number one and top seed aside 10-5 10-2.
"I am really pleased with the way I played and, to be fair, you have to play that way to beat Nicky," said Anderson.
"If Nicky got a wee break here or there, it might easily have gone the other way.
"Darren Burnett gave me a loan of his bowls, so I think that made the change."
Anderson's compatriot was not so lucky in his own match, losing to Gillett 8-4 8-4.
Brett had survived a scare in the previous round, when he was taken to a tie break by 71-year-old Scottish qualifier Charlie Bence, and was full of praise for ninth seed Anderson.
"Stewart was great today and I was chasing the shots you don't want to be chasing," he said.
"I would have liked to have played a little bit better myself, but Stewart didn't let me."
Anderson will face fourth seed Harlow after the Englishman claimed a last-bowl victory over former champion Gourlay, the fifth seed, to win 5-8 10-4 2-1.
It was a similar tight finish for Foster as he defeated 14th-seeded Englishman Jamie Chestney 6-10 10-4 2-1.
The Scot, who last won the title in 2011 and is seeded six, said: "Jamie deserved to win the first set and then, in the second set, you've got to dig in and then I lost a three at the first end.
"But you've just got to keep plugging away and I know today I was not at my best, but I showed a lot of fight and I know there's a lot of room in the tank yet.
"I dug deep and I am happy to get through. Somebody up there must like me."
Gillett, who defeated six-time champion Alex Marshall in the second round, continued his fine form to beat another Scot.
Burnett, the 2014 champion, Commonwealth Games gold medallist and seventh seed, was swept aside by the 16th seed who is seeking his first world title.
Scotland's Stewart Anderson knocked holder Nick Brett out of the World Indoor Bowls Championships to move into the semi-finals in Great Yarmouth.