The Czechs took a 1-0 lead at Juventus Stadium courtesy of a Libor Kozak strike, as they looked to strengthen their hopes of a top-two finish in Group B.
But two goals to the Italians early in the second-half guaranteed qualification for the hosts and all but extinguished the Czechs' dream of going to Brazil next year.
And the defeat had deeper consequences, as Bilek, 48, offered his resignation after almost four years in the role.
Interestingly, Bilek was quoted as saying he had offered to vacate the top job due to the nation's inability to qualify for the World Cup finals - despite their hopes having not yet been mathematically dashed.
The Czechs are four points outside second spot in the group with two matches to play.
Bilek was also quoted as saying whether he stays on for their final qualifying matches 'depends on the president'.
