It is his 13th test century, just four behind his mentor Martin Crowe.
Only McCullum, with his triple ton against India in Wellington last year, and Crowe's 299 against Sri Lanka, also at the Basin Reserve in 1990-91, have had higher test innings than Taylor.
At lunch Australia, trailing by 65 runs, were none without loss off one over.
Taylor scored at a brisk rate throughout, until joined by last man Trent Boult, who finished unbeaten on 23.
Taylor inched his way past England's Tip Foster's mark of 287 as the highest score by an overseas player in Australia, as the hosts spread the field to slow things down.
He had decent support through the latter stages of his innings from Mark Craig (40 for the seventh wicket), Matt Henry (29 for the eighth), Tim Southee (33 for the ninth) and finally Trent Boult, who shared a 37-run stand for the final wicket.
Boult helped himself to six, four, four in successive deliveries from seamer Josh Hazlewood late on as salt was rubbed into Australian wounds.
Taylor had a poor first test at the Gabba, being dismissed for 0 and 26. However he worked hard in the interim and had a rich reward today.
''After the way I batted in Brisbane I was pretty rusty and all over the show," Taylor said before play today.
''I trained the house down leading into that game, and obviously the results didn't show.
''I came into this game not putting too much expectation on myself, had a net for half an hour and it was nice not to think about it too much. It was nice to go out and express myself.
''(Brisbane) played a little bit on my mind. You start questioning yourself and little doubts creep in so it was nice to get rid of those."
NZ highest test totals v Australia
624, Perth, 2015
553-7 dec, Brisbane 1985
534-9 dec, Perth, 2000-01
Highest individual score against Australia in Australia:
290: Ross Taylor (NZ), Perth 2015
287: Tip Foster (England), Sydney 1903-04
277: Brian Lara (West Indies), Sydney, 1993
251: Walter Hammond (England), Sydney, 1928
241 not out: Sachin Tendulkar (India), Sydney 2004
New Zealand's highest test scores:
302: Brendon McCullum, v India, Wellington 2014
299: Martin Crowe, v Sri Lanka, Wellington 1990-91
290: Ross Taylor, v Australia, Perth 2015
274 not out: Stephen Fleming, v Sri Lanka, Colombo, 2002-03
267 not out: Bryan Young, v Sri Lanka, Dunedin, 1996-97
262: Fleming, v South Africa, Cape Town, 2005
259: Glenn Turner, v West Indies, Guyana, 1972
242 not out: Kane Williamson, v Sri Lanka, 2014
239: Graham Dowling, v India, Christchurch, 1968
230 not out: Bert Sutcliffe, v India, New Delhi, 1955-56