Centre Steve Matai was lucky to escape with a warning after going high on Paul Gallen, although Inglis went over in the left corner from the resulting penalty.
Thurston added the extras and it was against the run of play when New Zealand scored their first points on 26 minutes, with second rower Sika Manu stepping off his right foot to touch down on his debut.
Matai added the conversion to narrow the gap to just two points but Australia quickly restored their advantage when full-back Slater put Monaghan over in the corner. Thurston's third goal made it 14-6 at the interval.
Tate went off with a neck injury 10 minutes after the restart, suggesting his World Cup might be over.
Stuart promises more after Aussie win
Soon after, Thurston bamboozled the tiring New Zealand defenders with a mazy run, and a fine pass out of the tackle sent Slater over for a try. Thurston converted his fourth goal from five attempts.
Australia's Cameron Smith and New Zealand's Jerome Ropati both had late tries disallowed by the video referee before Folau burst over for his second just before the end, with Smith adding the touchline conversion.
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