With Burns on 27 and Khawaja 19, Australia were 70 for 1 at stumps requiring a further 131 at Hagley Oval here to sweep the series and reclaim the world number one ranking.
Jackson Bird, with his first five-wicket Test haul, and James Pattinson combined to end the New Zealand second innings at 335, giving Australia a 201-run target.
David Warner, who has struggled on New Zealand surfaces, was the Australian wicket to fall, caught behind for 22.
Burns offered chances but luck was on his side. On 19, he pulled Henry through square leg where Wagner was unable to hold a difficult chance. On 23, an inside edge off Matt Henry sailed close to off stump.
Khawaja, however, looked untroubled and effortlessly moved to 19 off 23 deliveries.
Although Australia have been dominant for much of the Test, New Zealand put up a valiant fight with two century partnerships as they battled to save the contest.
The loss of Anderson for 40 was the start of a Bird strike that claimed three wickets in six balls. Williamson and Tim Southee also fell before BJ Watling and Henry mounted rearguard resistance.
They added 118 for the eighth wicket to get New Zealand up to 328 for 7 before another batting meltdown saw the final three wickets fall for seven runs.
When Bird took the new ball in his next over he bagged the wicket Australia had fought all morning for bowling Williamson three runs short of what would have been a 14th Test century and two balls later removed Southee for a duck.
After Australia sought a review of umpire Ranmore Martinesz's not out decision and the third umpire concurred there was bat on ball, the Australians converged on the umpires.
The stumps microphone picked up an obscenity from Hazlewood as the players made their feelings known.
Where Williamson and Anderson had been the epitome of caution, Watling and Henry took a more aggressive route to try to disrupt the line of the bowlers who were finding some reverse swing.
For Australia, Bird took five for 59 and Pattinson four for 77.
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