In reply to Australia's imposing 559 for nine declared after winning the toss, the Kiwis were dismissed for 624 just before lunch, their highest ever total against their trans-Tasman rivals.
At tea, Australia were 128 for two in their second innings, an overall lead of 63, with captain Steve Smith on 61 and Adam Voges on 43.
Kiwi hopes were raised when Australian openers Joe Burns (0) and David Warner (24) fell early, leaving the home side precariously placed at 46 for two, with the injured Usman Khawaja (hamstring) not taking his place at number three.
The Kiwis were still 17 runs ahead at that stage, but Smith and Voges then added 82 runs at a rapid rate to steady the Australian innings in a match dominated by the bat and which looks destined to finish in a draw.
However, on 290 the right-hander holed out to substitute fielder Jonathan Wells from the bowling of spinner Nathan Lyon to end a 37 run 10th wicket stand with Trent Boult (23 not out).
Former skipper Taylor passed several milestones in his innings, which was his first century since November last year, and ended a form slump that even he admitted was starting to worry him.
He broke through 5,000 Test runs and posted the highest ever score by a visiting player in a Test in Australia, as well as becoming the first Kiwi to pass 200 against Australia.

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