New Zealand 179 and 111 for 3 (Ravindra 56*, Mitchell 12*) need 258 more runs to beat Australia 383 and 164 (Lyon 41, Phillips 5-45, Henry 3-36)
Nathan Lyon outfoxed Kane Williamson and menaced on a sharp turning Basin Reserve surface as Australia remained in the box seat for a first Test victory despite a stirring fightback from New Zealand on day three.
Chasing 369, New Zealand reached stumps at 111 for 3 with Rachin Ravindra and Daryl Mitchell weathering challenging conditions in an unbroken half-century partnership. They defied Lyon, who was the predictable danger bowler and took 2 for 27 from 16 overs.
Ravindra mixed defence with proactiveness like when he clobbered the offspin of Travis Head for a towering six late in the day's play. He reached his fifty off 77 balls by whacking a short delivery from Mitchell Starc to the leg-side boundary.
Ravindra did endure several anxious moments, including an lbw appeal from Lyon that Australia unsuccessfully reviewed, but made it through to give New Zealand hope of conjuring a remarkable victory. Given the difficult situation, Mitchell curtailed his innate aggressiveness to finish 12 not out from 63 balls.
New Zealand will need to rewrite the record books if they are to take the lead in this series, with the highest ever run chase at Basin Reserve being 277 for 3 by Pakistan in 2003.
New Zealand's comeback started when Australia lost 6 for 37 as Glenn Phillips became the first New Zealand spinner to claim five wickets at home since Jeetan Patel in 2008. He finished with 5 for 45 from 16 overs as Australia was bowled out for 164 in their sixth lowest total against New Zealand in Test cricket.
But New Zealand's chase started poorly when opener Tom Latham gifted Lyon a wicket when he nicked off a short and wide delivery on the stroke of tea. Lyon, who entered the attack in the sixth over, had a huge caught behind appeal turned down on Williamson's first delivery and Australia unsuccessfully reviewed.
The riveting battle continued after the interval with Williamson, who made three hundreds in four innings against South Africa, determined to make amends after his horrendous run out for a duck in New Zealand's first innings.
Williamson made a statement by climbing into a couple of rare short deliveries from Lyon, who reverted to around the wicket and was armed with a leg slip. Williamson fell in the trap, unable to get on top of a Lyon delivery that pitched on middle and straightened as he inside edged into the safe hands of Steven Smith at leg slip.
After a rare double failure, Williamson was visibly annoyed with himself as his modest career mark against Australia fell to 37.26 compared to 55.25 overall.
With spin spitting off the surface, skipper Pat Cummins turned to Head and it proved an inspired decision with his third delivery accounting for opener Will Young, who played needlessly and edged to slip where Smith completed a stunning one-handed catch. It moved Smith past Mark Waugh's tally of 181 Test catches and into sixth spot on the all-time list.
Lyon's performance had Australia remaining confident despite a collapse in the middle session that loosened their stranglehold.
As they seek a rare Test victory over Australia, New Zealand have seemingly been overawed at times in the series-opener. But Phillips has been a standout after he top-scored for New Zealand with a defiant 71 off 70 in their disappointing first innings of 179.
He has also stood up with the ball after frontline spinner Mitchell Santner was overlooked for this match. Having had Usman Khawaja stumped in the first session, Phillips provided New Zealand with a much needed spark shortly after lunch with the wickets of Head and in-form Mitchell Marsh on consecutive deliveries.
Head, who had made just one run in his last three innings, raced to 29 off 36 balls before holing out to long-off. Marsh was dismissed for a golden duck after being caught at short-leg with Phillips equalling his career best of four wickets in an innings.
Phillips soon captured his fifth after Cameron Green, backing up from his masterful unbeaten 174 in the first innings, poked to short-leg to end his 80-ball 34 with Young taking a very sharp catch.
Phillips was denied a sixth when Cummins was dropped twice, but Matt Henry claimed the last two wickets - finishing with eight for the match - to complete New Zealand's fightback.
Having never taken a five-wicket haul in first-class cricket, Phillips became the first New Zealand spinner to reach the feat at Basin Reserve since 2006 when Daniel Vettori, who is now part of Australia's coaching staff, claimed 7 for 130 against Sri Lanka.
A New Zealand rally looked unlikely when Lyon, the nightwatcher, dominated the first 30 minutes of play. Having scored the most runs in Test history without a half-century, with a highest score of 47, Lyon fell short after making a breezy 41 off 46 balls in the top score of the innings.
Lyon's milestone bid ended after whipping Henry to a leaping Young at midwicket, but he was back in the thick of the action later in the day's play.
