Neil Wagner dismissed both batsmen in identical fashion in the final half-hour to give his side some late reward for a day of toil under a cloudless Christchurch sky.
"Obviously the two wickets at the end helped us and were quite crucial," Wagner told reporters. "A bit of luck for us that we've been trying to get all day and obviously got us back into a bit of a better position at the end of the day.
"Six wickets to get is still a long way, a lot of hard work to do and a bit of focus tomorrow on making that happen."
Burns, who had been dismissed early in his innings only to have the decision overturned on review, was well caught low down by Martin Guptill at square leg for 170, his highest test score and third century.
Smith also fell to a short ball in Wagner's next over for 138, his 14th test century, caught by the same fielder.
Adam Voges was on two with nightwatchman Nathan Lyon on four as the visitors ended the day 363 for four, having all but mowed down New Zealand's first innings 370.
Having won the opener in Wellington comprehensively, Australia need only draw in Christchurch to clinch the two-match series and take the number one ranking in test cricket.
New Zealand's only other success on a wicket offering little for the bowlers was when Usman Khawaja fell in the first hour for 24 off paceman Trent Boult.
Burns and Smith then combined in a 289-run third wicket partnership, a record for Australia against New Zealand.
Burns was given out caught behind on 35 but immediately asked for a review and television replays showed the Matt Henry delivery had brushed his forearm and not his gloves.
"I was confident I was not out. I didn't change anything the next ball I was facing ... it's very satisfying," Burns said of his century.
"I've done a lot of work with this tour coming up. It'll give me a lot of confidence going forward, you always want to be backing up individual performances."
(Editing by Ian Ransom/Amlan Chakraborty)
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