A second-half strike from Memphis Depay settled an electrifying Group B encounter at Porto Alegre's Beira-Rio Stadium after a stunning Tim Cahill volley and a Mile Jedinak penalty helped the Socceroos take a 2-1 lead.
Netherlands -- who thrashed reigning champions Spain 5-1 in their opening match -- are now in pole position to finish top of the group with six points from two games.
Van Persie will miss the next game against Chile though after picking up a second yellow card.
And Louis Van Gaal's side were given an almighty fright by their fired up Australian opponents, who now face almost certain elimination.
If Spain fail to beat Chile in Group B's other game in Rio later Tuesday, Australia will become the first team to be eliminated from the 2014 finals.
But the Australians have enhanced their reputation with a gutsy performance in their 3-1 opening defeat to Chile and the heroic display against one of football's established nations.
After soaking up early pressure, Bayern Munich star Robben settled Dutch nerves with a superb solo effort on 20 minutes.
But Cahill equalised moments later with a stunning volley which crashed off the underside of the crossbar and into the net.
Australia then took a shock lead nine minutes after half-time, Jedinak drilling the spot-kick after Daryl Janmaat was adjudged to have handled.
Van Persie pounced four minutes later to make it 2-2 before Depay lashed in his 68th-minute winner for the Dutch, who were watched by Holland's King Willem-Alexander and his wife Maxima.
Spain were to fight for their World Cup place against the Chileans in Rio's Maracana Stadium.
Spain coach Vicente del Bosque dropped a bombshell with his team line-up shortly before the 4pm (1900 GMT) kick-off, announcing that long-serving stalwarts Xavi and Gerard Pique had been dropped.
Following their traumatic drubbing by the Netherlands last Friday, Spain will be eliminated if they lose.
Only three defending champions have previously fallen at the first hurdle -- Brazil in 1966, France in 2002, and Italy in 2010.
No defending champions have ever been eliminated after only two games of the first phase, putting Spain at risk of the earliest exit in history.
"Chile are a difficult team because we'll have to defend and we'll have to fight one against one," said Spain midfielder Andres Iniesta, scorer of the decisive extra-time goal against Holland in the 2010 final.
"We must be able to make the right passes, and look for those in space, so we can score goals."
In Group A, Cameroon and Croatia, beaten by Mexico and Brazil respectively in their openers, meet in Manaus where defeat for either side will mean the end of their last-16 hopes.
Croatia are still bitter from their opening loss to Brazil in which they had a goal controversially disallowed while the hosts won a dubious penalty.
Meanwhile, Mexico goalkeeper Guillermo Ochoa basked in plaudits after a string of fine saves ensured a 0-0 draw against Brazil on Tuesday.
"It was the best game of my life," said Ochoa, 28, after his stunning stops frustrated Brazil at Fortaleza.
One stop in the first half, down low to his right to keep out Neymar's header, had shades of a legendary save by England's Gordon Banks to deny Pele in the 1970 World Cup in Mexico.
The draw was greeted with howls of anguish by Brazil's media, with the Folha de Sao Paulo newspaper describing the performance as a "horror show."
"Brazil played badly. It's their worst start since 1978 and sows doubt as to their ability to face the big (teams) at the World Cup," the paper lamented. (AFP)