The Kangaroos downed the reigning World Cup champions three tries to two a year out from the next World Cup tournament in England.
Australia broke open a 10-all deadlock to prove superior to New Zealand in Townsville's tropical heat and clinched the Test match with a converted try by winger Darius Boyd nine minutes from time.
It was Australia's fifth straight victory over New Zealand since they lost the final of the 2010 Four Nations tournament to the Kiwis 16-12 in Brisbane.
Thurston's first of three goals brought up his 100th Test goal and he was at the heart of Australia's attacks along with scrum-half Cooper Cronk.
The Kiwis grabbed a seventh-minute lead when hooker Issac Luke ploughed over for a try from dummy-half, before his Australian counterpart and skipper Cameron Smith speared through an opening to score 16 minutes later.
The Australians looked to be getting on top when prop James Tamou, playing on his home club ground, charged over in the 34th minute off a delayed Cronk pass.
But New Zealand levelled at 10-all when winger Gerard Beale sprinted over in the corner after Sam Perrett had out-jumped Billy Slater to gather a high kick and trigger a passing rush to Beale's left wing.
Thurston was held up over the try-line midway through the second half as the Kangaroos upped the ante.
Thurston kicked the Australians in front 12-10 with a 62nd minute penalty after Krisnan Inu was penalised for stripping the ball off Greg Inglis in a tackle.
But Boyd scored the clinching try in the 71st minute after interchange player Robbie Farah got his pass away off the ground for Thurston and Paul Gallen to combine to send the winger racing away to score under the crossbar.