Labuschagne and Travis Head were unbeaten on 110 and 20 runs respectively after the home side elected to bat first in 40-degree (104 F) temperatures.
"It's great to get us in a position where we can kick on," Labuschagne said.
Coming in at the fall of opener Joe Burns' wicket after about an hour, Labuschagne, the newest batting find during the Ashes in England and against Pakistan, batted with huge confidence against a disciplined New Zealand attack that was diminished from the second session by the loss of injured fast bowler Lockie Ferguson.
Labuschagne danced down the pitch and hoisted left-arm spinner Mitchell Santner over the long on boundary for six to bring up his third straight century in his 12th test. He scored 185 and 162 in the tests against Pakistan recently.
Also in his innings, he reached 1,000 career test runs, and remained the most prolific run-scorer for Australia in 2019 with 939 runs (average 72.23) in nine tests.
Since being called up as a concussion substitute for Steve Smith during the Ashes, Labuschagne has grown in stature and matured into a fine right-hander occupying the pivotal No.3 spot.
"(Scoring three hundreds is) something I wanted to do and probably more of a dream," Labuschagne said. "For it to become a reality it's a different look.
"You don't want to take any runs for granted. When you get in you want to make a big score. I enjoy batting at No.3 (because) sitting and watching is draining."
He has batted for 318 minutes and faced 202 balls, and hit a six and 14 boundaries in his chanceless innings.
Labuschagne and Steve Smith shared 132 runs for the third wicket to see Australia through a period of uncertainty as New Zealand made the hosts work hard for runs after lunch.
While Labuschagne was scoring at will, Smith was uncharacteristically restrained and at times struggled. He took a blow on his fingers as he fended a short delivery from Neil Wagner, and on 19 was dropped by second slip Tom Latham off debutant fast bowler Ferguson.
Ferguson left the field midway through the day for a scan on a strained right calf.
Smith eventually fell for 43 when a Wagner short ball hurried on him and gloved an attempted pull to Tim Southee at leg slip — a well set trap. He faced 164 balls and hit four boundaries in his 188-minute innings.
Taking the second new ball, swing bowler Southee struck immediately when Matthew Wade shouldered arms to a delivery that swung and bowled the left-hander for 12.
Australia lost openers Burns and David Warner before lunch.
Wagner took a brilliant catch off his own bowling to remove Warner for just 43.
Warner was coming off back-to-back centuries against Pakistan, including an unbeaten 335 not out in Adelaide 10 days ago. He stroked four boundaries and faced 74 balls before hitting a full toss back to Wagner, who took a sharp one-handed catch to his right.
On a pitch that offered little assistance to the New Zealand bowlers, Warner and Burns shared 40 runs in the first hour until swing bowler Colin de Grandhomme struck.
Burns, on 9, was adjudged lbw by umpire Aleem Dar, who was officiating in a record 129th test match. Burns failed to review the decision as he hit across the line, but replays showed the delivery missing leg stump.
Australia is unbeaten in six day-night tests.