Tuesday's 1-0 loss away to Austria all but extinguished the Republic's hopes of qualifying for next year's World Cup finals in Brazil 2014 and that led the FAI to part company with veteran Italian manager Trapattoni and his assistant, Marco Tardelli.
"I want to thank everyone in Ireland who has given us their support during our time here which has always meant a lot to us," the 74-year-old Trapattoni said in a FAI statement.
"We leave this country with emotion because we understand the Irish supporters who have a well-deserved international reputation and they have our utmost respect."
He added: "I would like to thank (FAI chief executive) John Delaney, (president) Paddy McCaul, (honorary secretary) Michael Cody and the FAI board for their support and friendship over the last five-and-a-half years.
"I would also want to thank all FAI staff members, including the backroom team and the players, who have been great to work with during the last three campaigns.
"I wish them well in the future and hope that the job we have done leaves everything in a good place for my successor to take over," he added after the defeat by Austria left the Republic with only a slim mathematical chance of overhauling Sweden to take second place in European zone World Cup qualifying Group C.
Delaney said: "We thank Giovanni Trapattoni, Marco Tardelli and Franco Rossi for the last five-and-a half-years during which we qualified for our first major tournament in 10 years (Euro 2012) and were close to qualification for 2010 World Cup in South Africa after the play-off in France.
"This particular World Cup campaign has been disappointing, but Giovanni leaves us with a group of good young players which should form the basis of the squad that the new manager will use for the European Championships in France 2016 when 24 teams qualify."
Former Sunderland manager Martin O'Neill, himself an ex-Northern Ireland international, was installed by several bookmakers as the favourite to replace Trapattoni, with the FAI board due to meet in due course to consider their choice to succeed the Italian.
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