Gus Hiddink's reputation as one of the most astute coaches in international soccer looked in danger as his decision to drop Mark Schwarzer for Zeljko Kalac almost backfired when the keeper allowed a 56th-minute drive from Niko Kovac to bounce over him and into the goal to give Croatia a 2-1 lead.
However, a smooth strike by man of the match Harry Kewell, which followed Craig Moore's first half penalty, did the trick and propelled the Australians into the last 16 and history.
Australia now play highly-fancied Italy on Tuesday in sudden death.
The win gives Hiddink a chance to beat the Italians for the second successive finals. Four years ago he led South Korea to victory over Italy.
The European giants will go into the match with the Socceroos burdened by the match-fixing scandal which has dogged their country.
Juventus, AC Milan, Lazio and Fiorentina were yesterday charged in connection with the scandal and 13 of the 23 members of the Italian squad play for those clubs.
Hiddink, who will receive a $1.4 million bonus for getting Australia this far but will be off after the tournament to try to work his magic with Russia, paid tribute to the Australians' mental strength.
"I know this team has enormous mental power to be able to come back," said the 59-year-old Dutchman.
"I know this team, this team can react always and it did."
For once, though he got it wrong.
His decision to play Kalac - affectionately nicknamed "Spider" - was ill-conceived, and very nearly put the Socceroos out of the World Cup.
It was also unfair on Kalac.
Fate has delivered Kalac a difficult role for an elite sportsman - professional bench-warmer.
The giant keeper has been the perennial No 2 for his country for well over a decade, and fills the same spot for his Italian club AC Milan behind Brazilian first-choice Dida.
He has not started in a significant competitive match for Australia for years. In last month's pre-World Cup friendly against Greece in Melbourne he had very little to do, but still gave home fans the jitters every time he went for a high ball.
To suddenly throw him into the most important match in Australian soccer history was asking too much.
As captain Mark Viduka said: "It's difficult to come on for a game like that. I felt for him."
Kalac had a night in Stuttgart he would prefer to forget. Just before halftime he fumbled a corner so badly he very nearly allowed the ball to bounce over his own line.
And he will never want to see a videotape of his blunder which gifted Croatian captain Kovac his team's second goal.
Kalac could have done nothing to stop the second-minute free kick that Darijo Srna sent hurtling into the net; that was a top effort.
But the second ... Kalac summed it up best when he said: "I've taken a world class goal, and a shit goal."
It is difficult to understand Hiddink's reasoning for dumping regular goalie Schwarzer.
Schwarzer, a hero of last November's qualification penalty shoot-out against Uruguay in Sydney, had one or two nervous moments in the previous two World Cup matches against Japan and Brazil.
But he had done nothing to suggest he should get the chop.
Hiddink has pulled so many successful selection and tactical strings that he truly does deserve the description of magician and master coach.
But this time he blundered, and he was 11 minutes away from having to explain the inexplicable.
Kewell saved his blushes.
QUALIFIERS SO FAR
Germany
Ecuador
England
Argentina
Netherlands
Portugal
Brazil
Spain
Sweden
Mexico
Italy
Ghana
Australia
- REUTERS
Soccer: Coach wriggles out of Spider web
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