Louis van Gaal's men, who topped Group B, were staring at elimination in Sunday's last 16 clash until Wesley Sneijder drilled home in the 88th minute to cancel out Giovani dos Santos' superb opener.
But there was further drama as Robben - who has three goals so far in the competition - went down under a challenge from Mexico's veteran captain Rafael Marquez, leaving substitute Klaas-Jan Huntelaar to clinically dispatch the resulting spot-kick.
After the match, Mexico coach Miguel Herrera furiously claimed Robben "mugged" his team and accused him of diving on three occasions during the match.
The Bayern Munich winger did not entirely disagree, saying he went down too readily earlier in the contest.
But he insisted the award of the penalty itself, following rejected claims against Hector Moreno and Diego Reyes, was valid - Robben appearing to draw contact from Marquez before going to ground somewhat theatrically.
"I must apologise," he told NOS.
"The one (at the end) was a penalty, but the other one was a dive in the first half. I shouldn't be doing that."
On Moreno, who went off injured after tangling with Robben at the end of the first half, he added: "He hit me on my shin and must then (go) off himself with an injury.
"Then I get hit again if that is not a penalty, I do not know."
Robben ceded penalty taking duties to Huntelaar and spoke warmly of Netherlands' match-winner, who came on as a late substitute for Robin van Persie as the sapping Fortaleza heat took its toll on the Manchester United striker.
"Usually I am comfortable with taking a penalty, but this time I was the one who was tackled," Robben said.
"I felt comfortable that Huntelaar would capitalise.
"It was his moment, he deserved this. I'm almost crying from the emotions. The feeling is so strong, we want to win this, to fight together. These youngsters...I am so proud."
