Brendon McCullum's acrobatic effort on the field led to Shaun Marsh's dismissal
Lyon made most of the reprieve and played his shots.
Steve Smith led Australia's fight with a half-century
A superb 74-run stand for the ninth wicket helped Australia regain composure after losing their way and take a slender 22-run lead.
Mitchell Santner picked up his first Test wicket, clean bowling Josh Hazlewood.
Peter Nevill came to Australia's rescue with a late-order half-century that helped them eat into the deficit and eventually overcome it.
A massive controversy marred the second day's play when Nathan Lyon was given not out by the umpire inspite of HotSpot showing a nick.
Trent Boult ended Smith's recovery mission and sent him packing on 53
Josh Hazlewood of Australia celebrates getting the wicket of Ross Taylor of New Zealand during day two of the Third Test match between Australia and New Zealand at Adelaide Oval on November 28, 2015 in Adelaide, Australia.