The world's top-ranked side cruised to their victory target of 133 in less than 32 overs with a day to spare after bowling out New Zealand for 131 before tea.
The match had been in the balance at the start of the fourth day with the Kiwis 10 runs in front and all their second-innings wickets intact.
New Zealand, however, lost five wickets before lunch and the remaining five in the middle session as Australia seized control after trailing for most of the first three days.
"The boys are pretty proud of what we've just done," Australia captain Ricky Ponting [ Images ] told reporters.
"Coming here today the game was even, to think that it's over and done with now is pretty remarkable."
The fast bowlers ripped through New Zealand's top order with Jason Gillespie capturing three for 38 and Glenn McGrath [ Images ] and Michael Kasprowicz bagging one wicket each.
Leg spinner Shane Warne [ Images ] then destroyed the tail, taking five for 39 from 14 overs to notch his 29th five-wicket haul in Tests and sail past 1,000 first-class wickets in his career.
Unlike the first innings when they piled up 433, New Zealand were more tentative second time round. Seven of their 10 batsmen fell leg before wicket, equalling the world record for a Test innings.
Four batsmen made at least 20 but none did lasting damage, Brendon McCullum [ Images ] top-scoring with 24.
"We let ourselves down," New Zealand captain Stephen Fleming [ Images ] said. "Day four got us again through poor batsmanship and lack of application.
"We need to take the game by the scruff of the neck. At the moment we pinch the scruff of the neck and let it go."
Australia suffered an early setback in their run chase when Matthew Hayden [ Images ] was dismissed for 15 from the last ball before tea, caught in close by Craig Cumming off left-arm spinner Daniel Vettori [ Images ].
But Hayden's departure had no effect on the result as opener Justin Langer and captain Ricky Ponting shared an unbroken partnership of 110 to steer their team home.
Langer made an unbeaten 72 in 85 balls, striking 12 fours and a six, while Ponting belted seven fours and a six to finish 47 not out.
"It is very satisfying, I think all the guys know what we've just achieved," Ponting said.
"It's not just another Test match win, it's something we've done in a slightly different way."
Australia vice-captain Adam Gilchrist [ Images ] was named man of the match after his 121 in the first innings changed the course of the Test.
The tourists were in trouble at 201 for six when Gilchrist strode to the crease but finished one run behind New Zealand's first innings after he and Simon Katich [ Images ] (118) added 212 for the seventh wicket.
The second Test in Wellington begins on Friday with the final match in Auckland starting eight days later.
New Zealand (1st innings): 433 (H.Marshall 146, G.McGrath 6-115)
Australia (1st innings): 432 (A.Gilchrist 121, S.Katich 118, D.Vettori 5-106
New Zealand (2nd innings): (overnight 9-0)
C.Cumming lbw Gillespie 7
S.Fleming lbw McGrath 17
H.Marshall b Warne 22
L.Vincent lbw Gillespie 4
N.Astle b Kasprowicz 21
C.McMillan c Katich b Warne 5
B.McCullum lbw Gillespie 24
D.Vettori lbw Warne 23
J.Franklin not out 5
I.O'Brien lbw Warne 0
C.Martin lbw Warne 0.
Extras: (b-1, lb-1, nb-1) 3
Total: (all out, 50 overs) 131
Fall of wickets: 1-20, 2-30, 3-34, 4-71, 5-78, 6-87, 7-121, 8-127, 9-131
Bowling: McGrath 14-7-19-1, Gillespie 12-2-38-3, Kasprowicz 10-3-33-1 (nb-1), Warne 14-3-39-5
Australia (2nd innings):
J.Langer not out 72
M.Hayden c Cumming b Vettori 15
R.Ponting not out 47.
Extras: (nb-1) 1
Total: (for one wicket, 31.3 overs) 135
Fall of wicket: 1-25
Bowling: Martin 8-0-27-0, Franklin 5-1-26-0 (nb-1), Vettori 13.3-0-55-1, O'Brien 5-0-27-0
Result: Australia won by nine wickets to lead series 1-0
Man of the match: Adam Gilchrist (Australia)
March 18-22, Wellington
March 26-30, Auckland