Finch emerged from his T20 slump with his first half-century in 27 innings, Glenn Maxwell turned on the pyrotechnics with some thrilling stroke play, and Ashton Agar bettered his own record for Australia's best T20I figures as they kept the series alive with a 64-run victory in the third game at an empty stadium in Wellington.
Riley Meredith made an eye-catching debut with his pace helping take crucial early wickets even as New Zealand initially managed to keep up with the steep asking rate, but couldn't maintain it to make a serious dip at the target.
In the first half, the foundation for Australia was laid by a stand of 83 between Finch and Josh Philippe, and the latter part of the innings was powered by Maxwell's 31-ball 70 of which 62 runs came in boundaries. The eight overs from 10 to 18 brought 105 runs before New Zealand pulled it back somewhat.
Martin Guptill and Devon Conway played well in the chase, but Agar took three wickets in the 13th over to end the contest on his way to a six-wicket haul - just the fourth bowler to achieve that feat in T20Is while also becoming the sixth to take multiple five-wicket hauls in the format.
Australia 4 for 208 (Maxwell 70, Finch 69, Philippe 43, Sodhi 2-32) beat New Zealand 144 (Guptill 43, Agar 6-30, Meredith 2-24) by 64 runs