Finch had nine birdies and a bogey to reach 14-under 202 on The Hills course. Australia's Steve Bowditch, the second-round leader, was second after a 71.
Australian veteran Peter Fowler, the 1993 New Zealand Open champion, shot a 10-under 62 to move into third place, four strokes back.
'This is a new position for me and I am unsure what it will be like because I haven't been there before. But I'll try to stick to my usual routines,' Finch said.
Bowditch held the lead at 11 under, but a triple bogey at the par-5 ninth cost him momentum. He rebounded to birdie the 12th, 13th and 14th holes.
A late-night swim in icy Lake Wakatipu, the snow-fed jewel of the nearby resort of Queenstown, prepared Fowler for the lowest round of his pro career. He plunged into the lake to cool his anger after a second-round 75.
'I swam 100 meters in the lake and that cooled me off a bit,' Fowler said.
'It's part of my recovery anyway. I usually have an ice bath, so it was probably a little bit easier than sitting in a bathtub.'
Fowler's round included eight birdies and an eagle. He set a tough benchmark low round for the Hills course, which is barely a year old and hosting a professional tournament for the first time.
The tournament is sanctioned by the Australasian and European tours.