New Zealand's innings began badly with Martin Guptill lbw to Mitchell Starc for one off five balls.
Fellow opener Tom Latham had a couple of close shaves - and would have been lbw to Starc at 11 had Australia sought a referral.
But he dug deep and got to 36 before edging a catch to slip from offspinner Nathan Lyon, having added 81 with Williamson.
Taylor produced some nice drives and had got through the hard part to set himself up for tomorrow.
Earlier, Australia had added 143 runs to their overnight 416 for two, and lost seven wickets.
David Warner went early for 253, adding nine to his score at the start of the day; and captain Steve Smith pegged down by a far tidier bowling performance, had a slog to be out for 27.
Western Australians Adam Voges and Mitchell Marsh put on 50 before a late flurry of wickets gave some satisfaction to the New Zealand bowlers.
Offspinner Mark Craig finished with three for 123, while seamers Doug Bracewell, Matt Henry and Trent Boult got two apiece.
- By David Leggat in Perth