Both sides had the winning of the game, but the All Blacks were unable to get the ball over the line in the final stages while Australian replacement Reece Hodge's 57m, 82nd minute penalty goal attempt hit the post and rebounded into play.
The All Blacks looked to keep the game alive to seek the win and went so close to pulling the game out of the fire.
It set up the end of such a frustrating year dominated by the Covid-19 pandemic for a superb finale in the three remaining games to decide who will hold the Bledisloe Cup at the end of it all.
The sides meet again in Auckland next week and plenty will have been taken from the first exposure to the new coaching regimes of both sides and the new players who have joined the sides.
New Zealand should have had a handy lead at halftime but a disallowed try on the break kept the Australians in touch and they made the most of it.
Centre Rieko Ioane should have scored but he had tempted fate with his touchdown technique and a stunning breakout try was ruled out. Australia had attempted a lineout move back inside New Zealand's 22m area.
But the move became untidy when passes didn't meet their targets. Flanker Sam Cane secured the ball and it was slipped to Barrett and then released to the backline where Ioane was on hand to race for the line and a seeming try.
However, he failed to put the down and the try was ruled out.
Ironically, it was New Zealand's ability to execute a planned move, almost the same as Australia had tried in the first half, broke off the back of the lineout with hooker Codie Taylor running around the back, passing inside to wing George Bridge.
He raced into open space and then lobbed an overhead pass to halfback Aaron Smith. He fended off rival Nick White, who came back for a second attempt that unsettled, but not stopped, Smith who recovered to beat the defence to score.
When Australia did get their lineout sorted 52 minutes into the second half they pulled a set-piece play which allowed first five-eighths James O'Connor to double around and take advantage of defence absorbed inside to put left wing Marika Koroibete for a run into the corner for the Wallabies' first try.
Into the final quarter, a piece of opportunism at a breakdown where All Blacks fullback Damian McKenzie flicked the ball out of the tackle but it fell into unpatrolled space. White pounced and flicked a pass to right wing Filip Daugunu who went in at the corner in the 62nd minute to level the score at 13-13.
From the start of the game, Australia demonstrated powerful intent in taking the action to the All Blacks. Right wing Filipo Daugunu made some strong early bursts while they looked to get the ball into tighthead prop Taniela Tupou's hands as often as possible. He and lock Matt Philp were impressive on the burst and needed some tough defence to contain them.
However, failures at lineout time let them down badly in the first half.
They conceded the first try of the game after All Blacks flanker Shannon Frizell secured an Australian throw and in the resulting kicking play, Wallabies fullback Tom Banks' kick was fielded by McKenzie.
Ioane was alongside him to carry the play down the sideline, albeit with an undetected foot on the line, before the play was moved across field with lock Sam Whitelock featuring with a strong carry.
From the ruck, Frizell quickly handed the ball along to right wing Jordie Barrett and he ran in from 22 out to score eight minutes into the game.
Australia got their first points on the board when first five-eighths James O'Connor landed a 29th minute penalty goal.
Scorers: New Zealand 16 (Jordie Barrett, Aaron Smith tries: Barrett 2 pen) Australia 16 (Marika Koroibete, Filipo Daugunu tries; James O'Connor 2 pen). HT: 8-3
Tickets to the second Bledisloe Cup Test presented by Investec at Eden Park on Sunday 18 October are available HERE.