The Football Association of Ireland announced his departure on Wednesday after a meeting with Trapattoni and assistant boss Marco Tardelli.
The pair had been contracted until the end of May 2014.
In his five years in charge, Trapattoni guided the Republic to the finals of Euro 2012.
The 1-0 defeat in Austria left Ireland with only a mathematical possibility of overhauling Sweden to snatch second place in Group C.
"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," said Trapattoni.
"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."
Immediately after Tuesday's game in Vienna, the 74-year-old Italian had said he would wait for the FAI to decide on his future.
The verdict came the following morning with a statement announcing the manager's departure by mutual consent.
Trapattoni's Ireland record
Matches: 64
Wins: 26
Draws: 22
Defeats 16
Trapattoni's scheduled post-match news conference on Wednesday had already been cancelled by that point, a move which heightened speculation that his departure was imminent.
Speaking immediately after Tuesday's game, Republic captain Robbie Keane paid tribute to Trapattoni's efforts with the country.
"It is very difficult for a player just coming off a result and speaking about a manager who has done very, very well for this country," said an emotional Keane.
"He has been a credit to his country and he has carried himself the way a man of his stature should."
Keane added that there needed to be a "realism" about the Republic's capability at international level.
"We're a small nation and I think people get carried away sometimes and expect us to qualify all the time," he said.
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