The world number 12 will face Richard Gasquet for the title on Saturday after the French veteran was awarded a walkover in his semi-final against compatriot Constant Lestienne, who withdrew with a shoulder injury.
Norrie lost only a single point on his own serve in the opening set and was all over the American, finally achieving the breakthrough that all but assured him of the set after Brooksby had saved five break points.
Despite a wrist problem that needed an injury timeout, Brooksby made the second seed work hard for the victory in the second set but every time he asked a question, Norrie had an answer.
An early break of serve by the American was answered with an immediate break back to love and Norrie broke again to take a 4-3 lead, finally reaching his 12th career ATP final on his third match point.
"It was an absolute battle with Jenson, lot of long rallies, I know how well he competes so I'm just glad to get it done in straight sets," said Norrie, the runner-up in Auckland in 2019.
"I really enjoyed it. I'm glad the sun came out, everyone came out on a Friday, the atmosphere was great."
Britain is also represented in the last four of the other ATP warm-up for the Australian Open being played this week at the Adelaide International 2.
Jack Draper, who will play Rafael Nadal in the opening round at Melbourne Park next week, takes on South Korean Kwon Soon-woo in the first semi-final with local defending champion Thanasi Kokkinakis facing Spain's Roberto Bautista Agut in the second.
