The 74-year-old and assistant Marco Tardelli have both departed, officially by "mutual consent", in the wake of Tuesday's defeat in Austria which left the Republic's hopes of reaching the World Cup finals hanging by a thread.
Trapattoni was appointed manager in 2008 and came close to guiding the team to the 2010 World Cup finals, losing out to France in a tense qualifying play-off.
But the Republic qualified for last year's European Championships, where they scored only one goal in defeats to Spain, Italy and Croatia.
The Italian was awarded a new two-year contract by the FAI, but the team have struggled during the World Cup qualifying campaign and the lacklustre defeat in Vienna ensured they will not feature in Brazil next year.
In an FAI statement, Trapattoni said: "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.
"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.
"I would like to thank John Delaney, Paddy Mc Caul, 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."
FAI chief executive John Delaney added: "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 ten years 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."
The FAI will meet "in due course" to discuss candidates for the vacancy, with former Celtic, Sunderland and Aston Villa manager Martin O'Neill believed to be the early favourite for the role.
Sky sources understand Trapattoni has agreed to waive a portion of any compensation he is owed by the FAI.