Two tries in five minutes late in the first half from hooker Keven Mealamu helped the hosts to a half-time lead of 14-7.
Richie McCaw's try looked to have given the All Blacks a winning advantage, and although Scott Fava touched down to cut the deficit soon after, Australia's hopes were long gone by the time Isaia Toeava raced clear to seal a convincing victory win late on.
The decisive period came late in the first half when Wallabies number eight Rocky Elsom was sin-binned for repeatedly failing to roll away from the tackle.
That denied Australia the chance to build on their early breakthrough and the All Blacks took full advantage with Mealamu's unlikely double.
Tuqiri had opened the scoring in dynamic fashion in the 15th minute after some great lead-up work by full-back Chris Latham.
Latham fielded a kick inside his own 22 and as the defence approached, he chipped over the top and regathered before passing to Mat Rogers, who looked to get on the outside of Daniel Carter.
As the pair ran to the sideline, Rogers threw an overhead pass back to Tuqiri, who dived over for his 23rd Test try.
New Zealand fly-half Carter had a second unsuccessful penalty attempt in the 25th minute, but the All Blacks were handed the initiative soon after when Elsom was yellow-carded.
With a man advantage, the All Blacks spread the ball from wing to wing, and despite repelling a number of raids, Australia eventually crumbled when Mealamu dived on a loose ball from a line-out and scrambled over for a try.
Carter kicked the conversion to level the scores at 7-7 and the home team got their noses in front when Mealamu crashed over for his second try from a ruck close to the Australian line.
New Zealand went further ahead seven minutes into the second half when Carter slotted an easy penalty, and from the kick-off, Aaron Mauger made a break and chipped a kick deep into Australian territory.
Stephen Larkham fielded the ball on his line but fumbled in a challenge by Joe Rokocoko before being swarmed upon by the New Zealand loose forwards, and from the ruck, the TV match official correctly declared McCaw had forced a try.
Replays showed that the ball had become lodged between Larkham's legs and McCaw's was the only hand on the ball.
Australia hit back straight away with a try by Fava just 60 seconds after the replacement back-rower came on for his Test debut.
The Wallabies regathered from the kick-off and good forward play took them to the All Blacks' line, where following repeated rucks, Fava spotted a gap to dive over.
Outside centre Stirling Mortlock missed the conversion to leave the score 24-12, so when Carter kicked another penalty in the 58th minute, the All Blacks enjoyed a 15-point lead going into the final quarter of the game.
With four minutes remaining, Carter charged down Larkham's chip kick and passed to replacement back Toeava for a clear run to the line.
Australia will look to draw level next week in Brisbane when the trans-Tasman rivals face each other again in the second match of the series.