Centuries from Steve Smith and Adam Voges helped Australia build a 193-run lead after New Zealand's Ross Taylor had broken a 111-year record on day four of the second Test in Perth.
Taylor made 290, the highest score by an overseas batsman on Australian soil, as the tourists added 114 runs to their overnight tally to post a first-innings score of 624.
After finishing day three on 235 with the Black Caps on 510 for six, Taylor hit another 55 runs on Monday before he was the last man out, caught by substitute Jonathan Wells off the bowling of spinner Nathan Lyon.
Taylor struck 43 fours in his 374-ball innings as he passed the previous-best score by an overseas batsman in Australia, that of England's Tip Foster who made 287 at the Sydney Cricket Ground in 1903.
Lyon (three for 107) also claimed the wicket of the other overnight batsman, Mark Craig, while Mitchell Starc sent back Matt Henry and Tim Southee to finish with figures of four for 119.
Beginning their second innings 65 runs behind, Australia got off to a shaky start as Joe Burns went for a duck to Southee before his opening partner David Warner - who made 253 in the first innings - was caught by Tom Latham off Trent Boult for 24.
That left the hosts on 46 for two and still in arrears, but Australia captain Smith (131) and Voges (101) steadied the ship, sharing an unbroken stand of 212 as Australia reached 258 for two at stumps.
Australia lead the three-match series 1-0.