Australia made history in Perth, racking up the biggest points total ever scored against New Zealand as they scored six tries in an entertaining Rugby Championship encounter.
As a result, Wales will become the world's top-ranked rugby team if they beat England tomorrow in Twickenham.
Australia gave themselves a huge boost ahead of the World Cup when they stunned a New Zealand side reduced to 14 men for more than 40 minutes.
Lock Scott Barrett's dismissal for a dangerous tackle just before half-time undoubtedly had a big impact on the contest but the Wallabies were well worth their first win over the world champions since 2017.
Winger Reece Hodge scored two of their six tries as Australia ran up their highest tally of points in 165 tests against the All Blacks and gave themselves a chance of regaining the Bledisloe Cup they last raised aloft in 2002.
Australia, who play the second Bledisloe test in Auckland next week, could yet win the Rugby Championship for the first time since 2015 but they will need Argentina to beat South Africa in Salta later today.
That will matter little to former Leinster coach Michael Cheika and his team, who have endured four largely miserable seasons since they lost to the All Blacks in the last World Cup final.
Lock Lukhan Salakaia-Loto, scrumhalf Nic White, winger Marika Koroibete and fullback Kurtley Beale also crossed for the Wallabies in front of a packed house of 61,241 at Perth Stadium.
The All Blacks had tries from backs Anton Lienert-Brown, Rieko Ioane, Beauden Barrett and Ngani Laumape but they will not be happy with a defeat on the back of a draw against the Springboks.
Australia got off to a blistering start and led by 10 points after as many minutes.
Christian Lealiifano kicked the first points after Ardie Savea was penalised at the breakdown and Hodge got the first try after James O'Connor, starting his first test in six years, offloaded cleverly to set the winger free in space.
The All Blacks hit back with two tries in four minutes to take a 12-10 lead, the first when Lienert-Brown eventually touched down a Jack Goodhue kick through and the second when Ioane finished off a typically clinical move off turnover ball.
Lealiifano kicked his second penalty to put Australia back in front after 27 minutes and the match continued at a frantic pace with the home side playing some of their best rugby for the last four seasons.
They were pressing for another score just before the break when Barrett crashed into the head of Hooper with a no-arms tackle and became only the fourth All Black to be sent off in a test.
French referee Jerome Garces has been responsible for half of those dismissals having also shown a red card to Sonny Bill Williams for a shoulder charge during the second British and Irish Lions test in 2017.
Lealiifano kicked the penalty awarded to give Australia a 16-12 lead at halftime but nine minutes into the second half the shell-shocked All Blacks were down 26-12.
Salakaia-Loto forced his way over in the corner after 46 minutes before centre Samu Kerevi bulldozed down the touchline and got the ball away for White to cross for Australia's third try.
The world champions were not dazed for long, though, and finally got enough possession and pressure to give fullback Beauden Barrett the opportunity to dance his way through the defence and touch down.
Fly-half Richie Mo'unga added the extras to cut the deficit to 26-19 but Australia continued to press and winger Koroibete nicked the ball out of a tryline ruck to nip over and score a fourth try just after the hour mark.
Hodge crossed for his second after 69 minutes but Laumape replied almost immediately and it was not until Beale skipped over the line with a minute left on the clock that the Australians were able to relax.