Left armer Wagner, who had taken six wickets in the first innings, worked hard to unsettle the pair, but also spilled a sharp catch at square leg when Burns was on 16.
New Zealand scrapped hard today, largely on the back of two notable stands, having started the day at 121 for four, still 14 runs behind overall.
First Kane Williamson, with 97, and Corey Anderson, who got 40, put on a resolute 102 for the fifth wicket, batting through the entire first session.
However three wickets in 10 balls by surprise success story Jackson Bird undid that good work.
He got inside edges from both Williamson and Anderson, then had Tim Southee caught behind second ball.
Williamson felt three shy of what would have been his 14th test hundred, and put him in clear second spot on the New Zealand list behind only Martin Crowe's 17.
The second fine display came when wicketkeeper BJ Watling and seamer Matt Henry put on 118 for the eighth wicket with an enterprising stand.
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Watling followed his first innings 58 with 46 and Henry made his highest test score, 66, laced with 12 fours.
Bird finished with his first five-wicket haul in tests, five for 59, while James Pattinson took four for 77.
The low point of the innings came just before lunch when Australian bowler Josh Hazlewood was involved in an ugly situation after being denied an lbw appeal, then referral against Williamson.
Captain Steve Smith got involved and audible expletives were heard on the stump microphones. Hazlewood then appeared to give Anderson the benefit of his opinions on the incident. It is likely to cost Hazlewood part of his match fee.
WAGONWHEEL
MANHATTAN
