West Ham United have completed the signing of Ecuador striker Enner Valencia from Mexican side Pachuca.
Sky Sports understands Middlesbrough are lining up a move for Chelsea midfielder George Saville.
Egervori's men were well placed to qualify for the World Cup play-offs prior to the clash, starting the night in second, but they ended it wounded and in fourth.
A passage to Brazil is now all-but closed off to them and 63-year-old Egervori's offer to step down was accepted by the Hungarian Football Federation.
In a statement on its official website, the federation said: "The resignation was accepted immediately."
Egervori added: "It is fair that I offered my resignation. It was a real blow for Hungarian football and it was appropriate.
"I hope the hard work of the last three years will not be lost."
Egervori's assistant, Joseph Csabi, will take the team for the final World Cup qualifier against Andorra.