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Having reduced Australia to 113 for 6 after 13 overs chasing 220, Stoinis and Sams then got the equation down to 36 off the last three when Trent Boult produced a brilliant 18th over, conceding just six. It became 15 needed from the last over, the ball handed to Jimmy Neesham for the first time in the innings and he managed to remove both Sams and Stoinis to avoid the match slipping away.
The finish had echoes of Stoinis' magnificent 146 at Eden Park four years ago, after Australia were 67 for 6 chasing 287, and on that day it was a sharp piece of fielding from Williamson to run out last-man Josh Hazlewood that saved New Zealand.
"There were flashbacks of the last time Stoin hit us for a 100 or so, incredible partnership at the end and as we know on these sorts of grounds anything can happen," Williamson said at the post-match presentation. "You have to be on your game the whole innings because it can change so quickly. Think it was a pretty strong performance from us across the board but it's such a small margin in the T20 format.
"On these grounds, you are always in the game. So even though we had over 200 on the board, we know the power they have. We saw even when the run rate got out to 16, they were still well in the game and almost at a point where the game looked likely to go their way."
In the first half of the match, the most significant performance for New Zealand came from Martin Guptill who emerged from a lean season with 97 off 50 balls, getting to within a couple of metres of a third T20I hundred when he found Stoinis at long-off.
Guptill fell for a duck in the opening match in Christchurch but, on a surface that he said was a little tricky to start with until the initial shine had come off the new ball, he stroked the first ball of the match for four and launched the first of his eight sixes - among a combined tally of 31 for both teams - off Jhye Richardson at the start of the third over.
"Any first ball you get that is a wide half-volley you'd take, but it wasn't until about the third or fourth over that I started feeling a little bit at ease," Guptill said. "Kane came out around that time and we put on a great partnership to ease those nerves.
"It felt like I'd started batting well in the nets over the last little while then it hasn't quite gone out to the middle. Was nice to spend some time in the middle today and feel that rhythm back."
Despite Mitchell Santner's triple-wicket over leaving Australia in trouble, the closeness of the final result did not surprise Guptill. "Actually thought we were potentially ten runs short," he said. "Jimmy did a great job but was starved of strike in the last couple of overs. It ended up being enough... A great game in the end."