The rock-solid number three, last year's leading Test runscorer with 1104 runs at an average of 64.94, again proved the Black Caps' nemesis with his second ton of the series to continue his remarkable scoring sequence.
Supported by Steve Smith's 28th Test half-century, Australia reached stumps at 283 for three with Labuschagne unbeaten on 130 and Matthew Wade a breezy 22 not out.
The South African-born Labuschagne has now scored four centuries in seven Test innings this southern summer against Pakistan and New Zealand.
Smith, who took 39 balls to get off the mark, shared in a 156-run stand with Labuschagne before he again missed out after a lengthy stay and was caught at slip off Colin de Grandhomme for 63 off 182 balls.
David Warner fell again to the leg-side trap on the third ball after lunch when he was caught by de Grandhomme at leg gully off Neil Wagner for 45.
It was the fourth time in the series left-armer Wagner has snared Warner, who has yet to reach a half-century against New Zealand this summer after scoring an unbeaten 335 and 154 against Pakistan last November.
Opener Joe Burns was dismissed in the 15th over, squared up by de Grandhomme and caught by Taylor at first slip for 18.
De Grandhomme opened the bowling with Matt Henry after senior paceman Tim Southee was surprisingly left out of the attack.
Australia won the toss and chose to bat against the visitors, who made five changes to the team that lost the Boxing Day Test in Melbourne by 247 runs.
The Black Caps went into the Sydney Test without skipper and star batsman Kane Williamson, who has been ill in the lead-up.
Batsman Henry Nicholls and spinner Mitchell Santner were other illness withdrawals while Southee lost his place to leg-spinner Todd Astle.
Paceman Trent Boult is also missing after returning home with a broken hand suffered in the second Test in Melbourne.
Batsman Glenn Phillips, who made a late dash to Sydney as cover on Thursday, was named to make his Test debut, with opener Tom Latham to lead the side in Williamson's absence.
Will Somerville, Matt Henry and Jeet Raval were also called into the side, while the Australians were unchanged after considering Mitchell Swepson as a second spinner.
Australia have been unbeatable this season, winning all four Tests at home, two each against Pakistan and New Zealand, after retaining the Ashes in England.
The Test is being played against the backdrop of one of Australia's most devastating bushfire seasons with at least 18 people losing their lives in blazes raging across the country, including on the outskirts of Sydney.
Play will be suspended in the match at the umpire's discretion, should smoke significantly affect air quality or visibility, but the sky above the ground was clear during the first day's play.