Burns went lbw to left armer Trent Boult for 11; Smith speared a catch to second slip off Doug Bracewell at 35 before Smith was also lbw to Boult for 14.
New Zealand tails were up but they had no more success before the break.
The day began badly for New Zealand.
The object of exercise today was to bat as long as possible to enable New Zealand's bowlers to operate in the final, night session with a relatively new ball, where they get more help from the conditions.
They needed runs and time in the middle out of wicketkeeper BJ Wating but he departed in the second over of the session.
He pushed tentatively out at an outswinger from the outstanding Josh Hazlewood to be caught at second slipon, without adding to his overnight seven. He's had a poor series with the bat, managing just 83 runs at 16.6.
Mark Craig supported Mitchell Santner for a while before driving loosely at Hazlewood, giving wicketkeeper Peter Nevill the catch.
The best batting of the morning came from Santner, who grew in confidence through the morning, and Bracewell.
Slowly they pushed the lead up with a mix of the aggressive and the careful.
Santner cut and drove handsomely, and lofted offspinner Nathan Lyon for six over long on. However he was stumped next ball, trying to repeat the shot, having made the innings' top score, 45 of 88 balls.
Bracewell batted responsibly, also finding time to whip seamer Peter Siddle for six well over the square leg fence.
With the tea interval in sight, Tim Southee, having got to 13 in 16 balls, tried to swing seamer Mitchell Marsh over square leg and came up about 2m short. At that point New Zealand's lead was 153.
Racewell swung Marsh lustily to the mid wicket fence and New Zealand had 200 up.
However last man Trent Boult as bowled by star performer Hazlewood soon after
Hazlewood finished with best test figures of six for 70, while Marsh did well taking three for 59.
