The home side crumbled on the fourth morning in Wellington, losing their last seven wickets for 70 runs, with four of them falling to off-spinner Lyon. He finished with innings figures of 6-65 off 27 overs, giving him 10-108 for the match.
The 36-year-old capitalised on spin-friendly Basin Reserve conditions and some meek batting from New Zealand, who began the day on 111-3 and needing 258 more runs to secure an unlikely victory.
The capitulation began in the seventh over of the day when Lyon claimed the key wicket of Rachin Ravindra for 59. Ravindra had only added three to his overnight score when he failed to keep a cut shot down and was caught by Cameron Green at point.
Tom Blundell was caught in close by Travis Head without scoring in the same over and Glenn Phillips was trapped leg before by Lyon for one soon afterwards, to expose the tail.
Scott Kuggeleijn struck 26 off 28 balls before he failed to control a short ball from Green to be caught by wicketkeeper Alex Carey and Matt Henry fell for 14, caught at slip by Steve Smith off Josh Hazlewood.
Tim Southee became Lyon's sixth victim before Daryl Mitchell was the last wicket to fall, for 38, caught and bowled by Hazlewood.
Green was named man of the match for his career-high unbeaten 174 in the first innings.
No one else made three figures in a challenging Test for batsmen as Green rescued Australia from 89-4, having been put into bat, to reach 383 and set up a 204-run first-innings lead.
The second and final Test starts in Christchurch on Friday.
