Burns was on 27, while Khawaja was 19 not out with Australia 70 for one after being set 201 runs to win the match and series 2-0.
They had earlier dismissed the hosts in their second innings for 335 after tea.
David Warner was the only wicket to fall when he was caught down the leg side by BJ Watling from Neil Wagner for 22, though New Zealand needed to ask for a review to determine the ball had glanced his gloves.
Jackson Bird grabbed his first five-wicket haul to help wrap up New Zealand's second innings after tea that had been characterised by stubborn middle and lower-order resistance.
Bird, who had grabbed three quick wickets after lunch, including the key scalp of Kane Williamson (97), bowled Matt Henry for 66 -- his first half century and highest score -- then had Trent Boult caught for a duck to finish with 5-59.
"It was nice to take some wickets when the team needed it after lunch but I feel like I've been pretty inconsistent with the way I've bowled in the series which is disappointing," Bird told reporters. "But it was nice ...to put us in a pretty good position to win the game tomorrow and win the series."
James Pattinson finished with 4-77, but had the most important wicket in the final session on Tuesday when he dismissed Watling for 46 after the wicketkeeper had combined with Henry in a crucial 118-run partnership.
Until their eighth-wicket stand, the hosts had been in danger of being dismissed with a lead of only about 100 after Bird had broken a 102-run partnership between Williamson and Corey Anderson (40) that had got their side into credit.
The pair batted through a heated first session in which Australia dropped two catches, had two reviews turned down and an lbw decision overturned.
The second review, just before lunch and against Williamson, prompted a petulant outburst from captain Steve Smith and Josh Hazlewood, with the tall fast bowler likely to face further sanctions after microphones picked up his expletive-laden rant.
New Zealand had begun the day on 121 for four, 14 runs in arrears of Australia's 505, but their stubborn resistance ensured the match would enter a fifth and final day.
The target also appeared competitive though not overly challenging on a good batting surface and Australia should seal victory to reclaim the number one ranking in test cricket, though Anderson said his side would make it tough for the visitors.
"Any total you're going to have to chase in the fourth innings is always going to be a tough total regardless of what it is," the all-rounder said.
"It would have been nice to get a couple more poles (wickets) this evening. Hopefully tomorrow we can go bang bang and then potentially run through them."
(Editing by Peter Rutherford/Amlan Chakraborty)
