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Australia need 202 runs more with six wickets in hand at the end of day three of the second Test.
After 14 wickets fell on day one and seven more before tea, Williamson and Tom Latham had hit the pause button on a frenetic match with a partnership of 105 that erased Australia's first-innings lead and put the home side ahead.
Hazlewood's inexorable line and length, just on off stump and ducking away, was too much for the New Zealand top order.
The 32-year-old Conway will undergo surgery on his left thumb which got injured during the second T20I in Auckland on February 23.
The defeat leaves New Zealand with only one win in 24 Tests against their neighbours this century ahead of the second and final match in the series in Christchurch starting on Friday.
Wagner was picked in the squad for the two home Tests against Pat Cummins's Australia but was told by head coach Gary Stead he would not play in either match.
New Zealand consolidated their position at the top of the World Test Championship standings with 36 points and a winning percentage of 75.
India won by 214 runs after Musheer scored 131 to give a total of 295/8.
21-year-old Hana Rawhiti Maipi Clarke, who belongs to the country’s native Maori community, started her inaugural address with a ‘Maori Haka’ performance – a ceremonial war dance - pledging her commitment to her people.
Phillips was dropped on nought by Bangladesh skipper Najmul Hossain Shanto but maintained his composure after the Black Caps were reduced to 69/6 before tea to guide them home.