After being kept at bay by Tom Latham and Will Young for the first 90 minutes, the tourists picked up wickets at regular intervals.
Hazlewood finished with 5-31 in a masterly display of seam bowling while Mitchell Starc took 3-59.
It moved Starc up to fourth on the all-time Australian wicket-takers’ list with 357, overtaking Dennis Lillee, who ended his stellar career with 355.
Australia expected an early breakthrough when captain Pat Cummins elected to bowl in ripe conditions.
Dogged resistance from Latham and Young saw New Zealand reach 47 before Young was dismissed for 14.
From there, the New Zealand innings fell apart.
They lost eight wickets for 60 runs before the Australian onslaught was frustrated by a ninth-wicket partnership of 55 by Henry and Tim Southee in his 100th Test.
After Starc accounted for Young, Hazlewood bowled a precise line and length to claim the next four wickets including New Zealand talisman Kane Williamson, who was trapped in front for 17.
Williamson, who received a standing ovation as he arrived at the crease for his milestone 100th Test, sought a review but he knew it was a lost cause and started to walk before the three red lights were shown.
Hazlewood also accounted for Latham (38) and Daryl Mitchell (four), both caught by wicketkeeper Alex Carey as well as Ravindra caught by Usman Khawaja at first slip.
The impressive Carey took five catches including Tom Blundell (22) off Cameron Green and Glenn Phillips (two) off Starc, who then dismissed Scott Kuggeleijn lbw with his next delivery.
Hazlewood was also in on the action when he caught Southee (26) to give Cummins a wicket, then finished the innings in the next over when he had Henry (29) caught behind.
