FAI said that Irish supporters are among the best in the world and thanked them for the manner in which they greeted Euro 2012 elimination
Followers of the Boys in Green in Poland stayed behind in the stadium after the game, singing songs despite the heavy loss to La Roja.
The statement read: "The Football Association of Ireland would like pay tribute to our fans who have once again proven that they are amongst the very best in the world."
"The atmosphere that our supporters generated in the stadium was phenomenal and has brought great pride to our country.
"Our players and management are disappointed that they could not get the result tonight that would have kept us in contention to qualify for the knockout stages and give the fans what they are hoping for."
FAI chief executive, John Delaney, also echoed the warm sentiment, acknowledging the commitment of the fans to the team's cause.
"The Irish fans have made many sacrifices to follow our squad and have been absolutely amazing," said Delaney.
He added: "The abiding memory that we will take away from this match will be the many thousands of Irish fans singing the ‘Fields of Athenry’ right up to and beyond the final whistle.
"They are a credit to the game and to our country. On behalf of the FAI, the squad and management team I would like to thank them for their incredible support," he concluded.