That completed the offspinner’s 24th five-wicket bag and confirmed the Black Caps had for the second test in a row erred in opting for four seamers over a frontline spinner.
A couple of those seamers added some quick runs, Scott Kuggeleijn scoring 26 and Matt Henry 14, while at the opposite end Daryl Mitchell (38) played a lone fighting hand.
Lyon soon completed a 10-wicket haul for the test as the tourists sealed the match in a dominant first session.
New Zealand were made to pay for a shocking second day in which they allowed Australia a record 10th-wicket stand before collapsing to 29-5 in their reply.
Player of the match Cameron Green’s unbeaten 174 was the decisive innings in the test, helping Australia to the only total that exceeded 200 on a challenging pitch.
New Zealand are still without a home test victory over Australia since 1993, having beaten their transtasman neighbours only once in their last 32 meetings.
The Black Caps will have a few days to find answers before the second and final test starts at Hagley Oval in Christchurch on Friday.