Burns was 126 not out — poised to surpass his highest test score of 129 — and Smith was 79 unbeaten as Australia trailed by 129 runs in reply to New Zealand's first innings of 370.
Usman Khawaja was out for 24 to Trent Boult in the fourth over of the day and was the only wicket to fall on day two, as Burns and Smith comfortably held at bay New Zealand's four-pronged seam attack.
Their partnership had endured for 53 overs by tea.
Burns reached his century in 258 minutes, from 191 balls with 12 fours and with an edged four off Matt Henry which passed between slip and gully. Smith reached his 16th half century in 123 minutes, from 93 balls and with six boundaries.
In the most dramatic incident of the second day so far, Smith was felled by a bouncer from New Zealand fast bowler Neil Wagner three overs before tea. Smith tried to duck under the ball from Wagner which didn't bounce as high as he expected.
When he lost sight of the delivery, he turned his head away and the ball struck him on the back of the helmet. Smith first fell to his knees then slumped prostrate on the pitch, apparently dazed.
But he was able to get to his feet soon after and, responding quickly to treatment, was able to continue his innings with a new helmet.
New Zealand suffered for the lack of a specialist spin bowler. Expecting the pitch at Hagley Oval to stay green and moist for at least the first two innings, New Zealand named a four-pronged seam attack plus medium pace allrounder Corey Anderson.
The tactic backfired as the pitch became increasingly placid and New Zealand can only rely on Kane Williamson's part-time spin.
Australia won the first test by an innings and 52 runs and needs only a draw from this match to win the series and supplant India as the top-ranked test nation.
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