David Alaba's late winner consigned the Boys in Green to their second successive Group C defeat and all but ended their hopes of qualifying for the 2014 World Cup, forcing the FAI to review Trapattoni's position.
And, following a meeting between the association, Trapattoni and his assistant Marco Tardelli, a decision by 'mutual consent' was taken.
In a statement, the 74-year-old thanked everyone in Ireland for their support over the five and a half years in the job.
He said: "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.
Making the announcement, FAI Chief Executive John 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 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 Board of the FAI will meet in due course to discuss the process in relation to the appointment of a new manager.