Williamson was unbeaten on 70, the 25-year-old’s third successive 50-plus score in the series, after Australia had earlier declared their first innings at 559-9.
Ross Taylor (26) was batting at the other end with New Zealand 419 runs behind and still needing another 220 to avoid the ignominy of the follow-on.
Left-arm paceman Mitchell Starc trapped Martin Guptill lbw for one in the third over of New Zealand’s innings, while Tom Latham (36) fell to Nathan Lyon after raising 81 runs with Williamson.
Earlier, Australian opener David Warner was finally dismissed for 253 at the start of the opening session after adding just nine runs to his monumental overnight tally.
The hosts lost five wickets after lunch for the addition of just 73 runs as the Australian batsmen tried to build up the score ahead of the declaration.
Steve Smith also declared in both innings of the first test Brisbane, which Australia won by 208 runs to take a 1-0 lead in the three-match series.
New Zealand’s bowlers had managed just one maiden between them as Australia racked up 416 for two on Friday and the nine they notched up in the opening session were indicative of a far more disciplined effort.
Their first reward was the key wicket of Warner, who had savaged the tourists with two centuries in the first test at the Gabba and wreaked carnage with his maiden double century on Friday.
Clearly fatigued after nearly seven hours batting in the fierce Perth heat, the 29-year-old became the 12th Australian to score 250 in a test before being caught at second slip by Mark Craig off the bowling of Trent Boult.
Warner had faced 286 balls over 409 minutes in the 11th highest innings scored on Australian soil, notching 24 fours and two sixes and taking his tally for the series to 532 runs at an average of 177.33.
Captain Smith followed him back to dressing after scoring 27 runs from 91 balls when he tried to heave young paceman Matt Henry out of the ground but only managed to edge the ball to wicketkeeper BJ Watling.
That set the tone for a second session in which Australia’s middle and lower orders, deprived of batting in the series so far by the success of the top order, looked to ramp up the tally.
Western Australians Adam Voges (41) and Mitch Marsh (34) made the most of their chance, while the wickets of Peter Nevill (19), Mitchell Johnson (2) and Starc (0) added some respectability to off-spinner Craig’s figures (3-123).