Wagstaff finished the evening with 19 points, while Jean-Pierre Tokoto led all scorers with a 25-point haul and Lucas Walker featured with 11 rebounds and 12 points.
It's the first time the Wildcats have won in New Zealand since 2013 and followed two defeats at the Kiwis' hands in November.
Alex Pledger's 15 led the scoring for the Breakers, with Edgar Sosa and DJ Newbill next best on 10 apiece.
Perth started and finished the opening quarter on a high, Bryce Cotton nailing a three-pointer to kickstart proceedings and Clint Steindl sinking three free-throws with two seconds left on the clock.
Ahead 21-19 at the first break, the Wildcats attacked the second quarter with more energy than the Breakers, opening out to a seven-point lead with less than three minutes remaining.
Some sharp words from coach Paul Henare's time-out fired the Breakers into action.
Back-to-back treys from Sosa and Newbill narrowed the gap before Pledger's jump shot stole the lead back for the home team 36-35 at the half-time break.
Tokoto sparked a Perth revival midway through the third spell and the visitors opened out to a 55-44 lead on the back of an 11-0 scoring run.
Tokoto was everywhere - nailing free throws, grabbing rebounds, slipping assists and setting up his own slam dunk with a magnificent steal.
A late rally from the Breakers narrowed Perth's lead to 59-52 at the three-quarter mark but things didn't look good for the Kiwis, particularly after Rob Loe fouled out two minutes before the final break.
The Kiwis narrowed the gap to 61-59 two minutes into the last quarter but Wagstaff's deadly eye from long-range soon put the game back out of reach.
