In scorching conditions in Fortaleza on Sunday, Mexico looked set to reach the quarter-finals of the tournament for the first time since 1986 thanks to Giovani dos Santos' sweet left-footed shot from 25 yards three minutes after the interval.
Mexico held the lead for 40 minutes until substitute Klaas-Jan Huntelaar nodded Robben's corner into the path of Wesley Sneijder to lash home from just inside the area after 88 minutes.
In stoppage time, Robben - who had seen two penalty appeals already waved away by Portuguese referee Pedro Proenca - went down under a challenge from Rafael Marquez and the official pointed to the spot.
Up stepped Huntelaar to dispatch the spot-kick and send Netherlands into the last eight.
Mexico coach Herrera was less than impressed with Proenca's performance and insisted that Robben had dived to win the decisive penalty.
"It's a World Cup where everything was against Mexico from the start," he said.
"Robben threw himself to the floor three times and no-one said anything to him.
"After going 1-0 up the team retreated and that cost us. We gave opportunities to a team that had done nothing.
"Thank you for all the people who came to support. Unfortunately we could not get what we wanted."
