The tourists lead the series 1-0 and go into the final day requiring 131 more runs for victory with nine wickets remaining.
Joe Burns (27*) and Usman Khawaja (19*) are the unbeaten batsmen overnight with Australia 70-1.
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Joe Burns and Usman Khawaja had Australia in sight of victory after an absorbing fourth day of the second Test Tuesday marked by stubborn resistance from New Zealand and umpire abuse by the tourists.
With Burns on 27 and Khawaja 19, Australia were 70 for 1 at stumps requiring a further 131 at Hagley Oval in Christchurch 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.
Neil Wagner, using the short ball that provided him with six wickets in the first innings, caught Warner’s glove and BJ Watling completed the dismissal down the leg-side.
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.
Kane Williamson made 97 when adding 102 for the fifth wicket with Corey Anderson.
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.
Wagner putting in an inspired spell. He's split the webbing on his left hand so is bowling angry, short & fast. Interesting viewing. 54/1— BLACKCAPS (@BLACKCAPS) February 23, 2016
Anderson, having pocketed his usual attacking instincts, occupied the crease for three hours before shelving caution and driving at a wide delivery which he edged on to the stumps.
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.
Williamson had spent 340 minutes in the middle defying the Australians whose frustrations spilled over just before lunch with a Josh Hazlewood shout for lbw when Williamson was on 88.
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.
Henry, in his fourth Test, had 12 fours in his 66, easily surpassing his previous best 27.
For Australia, Bird took five for 59 and Pattinson four for 77.
From 157 for three in the 17th over, England were bowled out for 171 with two balls of their scheduled innings unused – after Jos Buttler (54) and Eoin Morgan had set course for far better.
De Villiers (71) then wasted no time putting England’s efforts into brutal context, with South Africa’s fastest-ever Twenty20 50 from 21 balls, as Morgan’s tourists lost a fifth successive international white-ball match in the space of 13 days.
De Villiers’ opening stand of 125 with Hashim Amla (69no) took only 50 balls, before he was caught in the deep off Adil Rashid – allowing South Africa to coast the remainder of the chase and still achieve it with a remarkable 5.2 overs to spare.
It is a godsend, in the circumstances, that England can fly home on Monday without any further battering of confidence before their ICC World Twenty20 campaign next month.
They lost their last seven wickets for 14 runs after Morgan and Buttler’s fourth-wicket stand of 96, as a flurry of mistimed shots and a fair slice of misfortune combined to undermine previous efforts.
Kyle Abbott (three for 26) was the principal beneficiary, and found himself on a hat-trick as England flailed away in vain.