Bayern Munich star Robben -- regularly accused of going to ground too easily throughout his career -- tumbled to the turf in injury time after making contact with Mexico defender Rafael Marquez.
Portuguese referee Pedro Proenca pointed to the spot and Klaas-Jan Huntelaar stepped up to bury the winning penalty.
It came just moments after Wesley Sneijder had scored an 88th-minute equaliser to cancel out Giovani dos Santos's superb goal for Mexico.
The epic finale settled a second round battle played in furnace-like conditions at Fortaleza's Castelao Stadium.
Mexico coach Miguel Herrera lashed out at referee Proenca for awarding the penalty.
"The determining factor was the man with the whistle. He put us of the World Cup," Herrera said.
"At the very least they can look at this and this referee ought to be going home like us."
Robben however insisted Proenca had been right to award the penalty, telling Dutch broadcaster NOS: "I was fouled."
But Robben did admit to a dive in an unsuccessful attempt to win a penalty earlier in the first half.
"I really have to say and at the same time apologise in the first half I took a dive and I really shouldn’t do that," Robben said.
"That was a stupid, stupid thing to do but sometimes you’re expecting to be struck and then they pull their leg away at the last minute."
- Second round jinx -
The Dutch fightback left Mexico's players in tears as their last 16 jinx struck again.
Mexico have now lost in the second round in six consecutive World Cup appearances, and had been just minutes away from a first quarter-final since 1986.
The match saw three-minute cooling breaks used for the first time in World Cup history as pitch-side temperatures hit 39 degrees (102.2 Fahrenheit).
Dutch coach Louis van Gaal praised his team for not wilting as they chased an equaliser in punishing conditions.
"The players showed they had faith and belief until the very end," said Van Gaal.
"The humidity was not in our favour so when you see that until the very last minute we were fresher and fitter than the Mexicans that is a big compliment to my players."
Van Gaal also revealed he had exploited the cooling breaks to introduce tactical tweaks at key points in the game.
"That is a clever way of benefiting from these breaks," he said.
The win puts the Netherlands into a last eight meeting with either Costa Rica or Greece in Salvador next Saturday.
The Costa Ricans played Greece in their last 16 meeting Sunday where the winner will reach the last eight for the first time.
Costa Rica sensationally beat Italy and Uruguay and drew with England as they topped Group D. Georgios Samaras scored a penalty in the third minute of injury time to give Greece the points they needed in their last Group C game to get the second qualifying place at the expense of the Ivory Coast.