The British No. 2, who grew up in Auckland and represented New Zealand until switching allegiance in 2013, hit 17 aces on his way to a 7-5, 4-6, 6-3 victory.
Norrie secured his place in the final by coming back from two double faults in a clutch final service game, characteristic of a match in which he saved eight of the 10 break points he faced.
"I was really, really nervous in that last game... slightly tight, but a big thanks to the crowd for getting me through that," Norrie said.
"It's so special... It's my first final and to do it at home, too, it means so much to me."
Norrie's success will likely be overshadowed by Andy Murray's retirement announcement, but he will be one of Britain's main hopes at the Australian Open next week when he opens his campaign against American Taylor Fritz.
Sandgren was in ominous form winning 31 of his 33 of his first serve points as he cruised to a comfortable 6-4, 6-2 win.
The 27-year-old will also be playing for his first ATP Tour title.