The inclement weather denied the Kiwis the chance at holding the trophy for a fourth consecutive series.
The tourists' hopes of securing victory in the three-match series now hinge on winning the final match on Saturday, when the weather is expected to fine up.
But success in Wednesday's series opener has left Mitch Marsh's side 1-0 ahead and unable to be overtaken by the Black Caps.
Play did not begin until 6.30pm AEST (9.30pm local time) on Friday - some two hours 15 minutes after the scheduled time.
Australia lost the toss and were made to bat first in a game condensed to nine overs per side.
A glorious six from Marsh (9 not out) over long on was the only highlight for Australia, who lost their other opener in Travis Head (5) in the second over.
Head miscued an in-swinger from Jacob Duffy (1-13) and found himself caught by a loping Devon Conway at backward point.
Rain forced the covers back on after the first ball of the third over with Australia 1-16 and 6.5 overs left in their innings.
The match was abandoned just before 7pm (AEST).
"Days like today are always difficult for both teams," said Marsh.
"You're waiting around all day. Obviously we do our best to try and get a game in, for the crowd especially, who hung around very patiently.
"But the weather won out today."
Australia had called Sean Abbott in for Ben Dwarshuis in the only change to the XI that won the series opener.
It comes as selectors deliberate which paceman will replace Mitch Starc in their frontline attack at next year's World Cup following the big left-armer's retirement from T20Is earlier this year.
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