With two minutes to play against the Mexicans in Sunday's second-round clash, Louis van Gaal's side were 1-0 down and heading for the exit.
But midfielder Wesley Sneijder pulled them level before Klaas-Jan Huntelaar crashed home their winner from the penalty spot in the 94th minute.
"Orange Miracle" ran the headline in the online edition of leading Dutch newspaper De Telegraaf. "The Dutch team stood on the brink of the abyss."
Several papers analysed the controversial Dutch penalty that gave them their winner after Arjen Robben went down in the box under a challenge from Mexican captain Rafael Marquez.
Marquez clearly stepped on Robben's toe as the Dutchman tried to wriggle past him but, not for the first time in the match or his career, Robben made the most of the incident by going down in dramatic style.
Robben later insisted it was a penalty although admitted to diving earlier in the match, describing his own behaviour on that occasion as "stupid".
The Dutch now face Costa Rica on Saturday in Salvador, scene of their 5-1 demolition of defending champions Spain in their opening match of the tournament.
And although they will be wary of the Central Americans, they will be overwhelming favourites to beat them.
Costa Rica, one of the biggest surprises of the tournament so far, had defender Oscar Duarte sent off and he will be suspended for the quarter-final. They will also probably be without fellow defender Roy Miller, who has a hamstring injury.
Their gruelling defeat of Greece in a penalty shootout on Sunday may also take its toll. They played for nearly an hour with 10 men and over the full match covered a remarkable 131 kilometres (80 miles) of ground between them.
Van Gaal's main injury worry is to key midfielder Nigel de Jong, who had to be substituted after just nine minutes against the Mexicans.
"It's a groin injury," Van Gaal said. "He is one of the most important elements in the team, so that was a big drawback, a big setback for us."
On the plus side, defender Bruno Martins Indi made a successful return for the Dutch having missed their previous match with concussion.
Robin van Persie had an unusually quiet match against the Mexicans and was substituted late on, but Van Gaal confirmed the striker was not injured.
"I took him off for tactical reasons," he said.
(Reporting By Gideon; editing by William Schomberg)
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