For the first time in the series, New Zealand are on top of their opponents at Hagley Oval and are well placed to score a rare victory over their rivals.
Australia will start day four on 77-4, chasing 279 for victory.
Travis Head is on 17 and Mitchell Marsh 27.
The highest successful fourth-innings run chase in a test at Hagley Oval was the 285 New Zealand scored to beat Sri Lanka with a run on the final ball of a dramatic match last year.
Australia spinner Nathan Lyon said there was a lot of confidence in the side that they can get up for what would be a great test win.
"There is a lot of belief in that changing room, and so there should be, but we understand that we're coming up against a class New Zealand bowling attack," Lyon told the media.
"Matt Henry has been the pick of the bowlers the whole series, so there's going to be plenty of respect there.
"But if we do good things for a long period of time then there is no reason why not."
Sunday was the best day New Zealand has had in the two match series so far. Getting through to 372 in their second innings and then reducing the Australian top order to 34 for four at one stage.
Matt Henry and Ben Sears have taken two wickets each in their second innings.
"If you'd said at the start of the day that we'd have them four down still 200 runs behind, I think we would have taken that," said Black Caps opener Tom Latham.
"The guys showed a lot of fight with the bat and we built a couple of key partnerships, which was great, and the guys did a great job tonight with the ball."
Latham said it would be a great result if they can get on top of Australia today, but he concedes they can't look too far ahead.
"It would be fantastic if we do but for us it is about taking it each ball at a time, I know it sounds a bit cliché.
"For us to build partnerships with the ball and if we can get some early wickets then hopefully we can do the job.
Latham said Head and Marsh are both attacking batters which may help them.
"Those two are guys that like to play on the front foot and can score quite quickly but on the other hand that presents opportunities."
New Zealand's last home test win over Australia was at Eden Park in 1993.
