The reigning T20 World Cup champions won with seven balls to spare on Sunday night, scoring 8-165 in reply to New Zealand's 2-162.
Number 10 Kim Garth faced eight balls without scoring before hitting the winning boundary.
It gave Australia a 1-0 lead in the three-match Rose Bowl series with the remaining matches at North Sydney Oval on Tuesday and Thursday.
Australia were charging towards victory when star veterans Beth Mooney and Perry had taken the score to 1-86 off just 8.2 overs, but the loss of Mooney for 38 from 25 balls in a terrible run-out mix-up changed the game as wickets started to tumble.
"There's probably an element of not being clinical tonight," said Perry, who top-scored for Australia with 52 from 30 balls.
"A lot of our batters haven't batted in the last 10 days. We went up to Brisbane and played three ODIs against Bangladesh and I think there were only about five girls who really had any significant time in the middle.
"There's an element of that, along with new conditions and a new format we haven't played since July."
Perry raced to her equal fastest T20 half century in just 27 balls but dragged a short ball from Jess Kerr to deep backward square, where she was well caught by Issy Sharp running to her right.
Georgia Wareham (10 from 7 balls) sliced a leg-spinner and Jess's younger sister, Amelia Kerr, straight to point.
Ash Gardner (23 from 17) gave Jess Kerr her third wicket, lbw on review, and Annabel Sutherland (nine from 5 balls) added a fourth pulling a short ball to deep square leg.
Nicola Carey (four from 3 balls) was run out by the recalled Tahlia McGrath and when captain Sophie Molineux (11 from 5 balls) was bowled Australia had lost 7-62.
The victory was Australia's 11th T20 win in a row, with their previous triumph having brought them World Cup glory at Lord's in July, while the White Ferns have now lost six of their past nine matches.
Not even a record partnership and a record-equalling total in Australia could secure New Zealand victory.
After a difficult start by the White Ferns, player-of-the-match Georgia Plimmer and Maddy Green combined for an unbeaten 127 in 80 balls. Plimmer made a career high 85 off 68 balls and Green 56 from 37 balls.
It was a breakout innings for the 22-year-old Plimmer, her fourth T20 half-century in 54 matches for New Zealand. Going into the game, she had an average of 16 and a strike rate of 104.
She was the beneficiary of two difficult chances, a stumping and catch from wicketkeeper Beth Mooney standing up to seamers Sutherland and Lucy Hamilton.
This is a new look White Ferns side following the retirements of key veterans Sophie Devine, Suzie Bates and Lea Tahuhu following a disappointing T20 World Cup campaign in the UK last June.
