Hungary had started the evening second in Group D and with genuine hopes of qualifying for next summer's World Cup.
However, the heavy defeat in Eindhoven, along with wins for Romania and Turkey elsewhere, severely dented their chances of booking a spot at Brazil 2014.
Egervori, 63, subsequently offered his resignation, which was swiftly accepted.
"It is fair that I offered my resignation. It was a real blow for Hungarian football and it was appropriate," Egervori told reporters.
"I hope the hard work of the last three years will not be lost."
Joseph Csabi, who was Egervori's assistant, will now take charge of the team for their final World Cup qualifier against Andorra.