After electing to bat first on Friday, the White Ferns posted 2-137 on the back of an unbeaten 90-run partnership between Amelia Kerr (51no) and Sophie Devine (39no).
In her first international back after taking a mental break in December, star veteran Devine impressed at the crease with Kerr.
Stand-in captain Tahlia McGrath and quick Darcie Brown took Australia's only wickets.
The tourists lost their way late in the innings, with substitute fielder Alana King dropping a catch.
There were other misfields that allowed New Zealand to continue moving their innings along.
In a major blow for Australia, Gardner has been sent for scans on her right index finger after leaving the field, replaced by King.
Given it is a T20, Gardner is unlikely to be able to complete her scans, get the all-clear, then return in time to bat.
Apart from being without injured captain Alyssa Healy, Australia picked a full-strength XI for the first of three games.
The White Ferns are the reigning T20 world champions after pulling off a surprise win at last October's tournament.
Australia, who had won the previous three T20 World Cups, were eliminated in the semi-finals and will be desperate to reassert their dominance.
Star bowler King, who was player of the series in the recent Ashes thumping of England, was left out in preference of fellow legspinner Georgia Wareham.
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