The off-spinner led the powerful Australian bowling attack to dismiss the Kiwis for 136 and seal another heavy win over the Black Caps after similar victories in Perth and Melbourne.
Australia have been unbeatable this season, winning all five Tests at home - two against Pakistan and three against New Zealand - after retaining the Ashes by drawing the series 2-2 in England.
Australia declared their second innings at 217 for two with David Warner scoring an unbeaten century, leaving the Black Caps with a revised 416-run target in the fourth innings on a wearing Sydney Cricket Ground pitch.
But the Kiwis buckled under the pressure of Australia's superior bowling attack with Mitchell Starc taking three for 25 to support the wiles of spinner Lyon.
New Zealand were reeling early at 27-4 and never recovered after Starc and Lyon took two wickets each in the middle session to put the skids under the tourists.
Starc removed both openers, Tom Latham and Tom Blundell, in the first five overs.
Blundell fell to a stunning catch by a diving Lyon at point for two and stand-in skipper Latham lost a review for leg before wicket on umpire's call for one in a shattering blow to New Zealand's hopes.
Jeet Raval was out in a review to the faintest of edges on 'Snicko' in Lyon's first over for 12, with the Kiwi innings unravelling under the onslaught from the Australian attack.
First-innings top-scorer Glenn Phillips went for a duck after technology detected a faint outside edge to wicketkeeper Tim Paine off Lyon leaving the Kiwis a shaky 22 for four.
Taylor's Kiwi record
Ross Taylor became the leading all-time Kiwi batsman, going past Stephen Fleming (7,172) before he was bowled by Pat Cummins for 22 to take his Test aggregate to 7,174.
Big-hitting Colin de Grandhomme smacked Lyon for six over extra cover to bring up his fifty but went next ball hoicking to Joe Burns at deep mid-wicket to take a comfortable catch for 52.
Todd Astle was out to a superb diving catch by James Pattinson in the outfield for 17 to give Lyon his fourth wicket of the innings late in the day.
Starc yorked William Somerville's middle stump for seven and BJ Watling was the last to fall, caught at backward square leg by Pat Cummins for 19.
Earlier, Warner completed his 24th Test century with a clip off his pads for three runs through mid-wicket and remained unbeaten when skipper Paine declared upon the dismissal of Marnus Labuschagne.
Labuschagne, who was dropped on four in a regulation caught and bowled chance by leg-spinner Astle, was caught at long on off Matt Henry for 59 - his seventh score over 50 in eight innings this golden domestic summer.
Labuschagne finished the home five-Test season against Pakistan and New Zealand with a stunning aggregate of 896 runs, made up of his 215 in the first innings, three other centuries and three half-centuries in eight innings.
There was drama late in the innings with the Australians chasing quick runs ahead of the declaration when Warner was given an official warning by umpire Aleem Dar for running down the middle of the pitch in scampering a single.
It resulted in five penalty runs being added to New Zealand's first innings total meaning their target was revised down from 421 to 416.
The Test was played against the backdrop of one of Australia's most devastating bushfire seasons with at least 24 people losing their lives in blazes raging across the country, including on the outskirts of Sydney.