France booked their place in next year's finals in South Africa with a 1-1 second-leg draw in Paris on Wednesday after striker Thierry Henry handled the ball in the build-up to their goal.
"The blatantly incorrect decision by the referee to award the goal has damaged the integrity of the sport and we now call on FIFA, as the world governing body for our sport, to organise for this match to be replayed," read an FAI statement.
The FAI issued the statement as Ireland manager Giovanni Trapattoni was telling a news conference it would be impossible to replay the match.
"I cannot ask this because I know it is impossible," said Trapattoni after his team had gone out 2-1 on aggregate. "When a referee decides, it ends the game for me."
The Italian instead criticised FIFA's late decision to seed the play-off draw and called on the ruling body to scrap extra time in similar games as it handed an advantage to the team drawn at home in the second leg.
"There are many doubts that remain that have to be eliminated. I will go to FIFA and advise them on what we think," said Trapattoni before adding video replays were likely to be brought in for the future.
FAIR PLAY
FAI chief executive John Delaney told a separate news conference that he had also written to the French FA on the matter and was not calling for a replay simply out of principle.
"When you ask me is this clutching at straws, we have to do what we are doing. We have to do it. It is up to the people who govern the game now, if they really believe in the principals of fair play then step forward," Delaney said.
"Everytime I go to congress it is all about fair play and fair play ambassadors... but well done is better than well said. They're words, I'd like to see actions," he said.
The FAI said there was a precedent for a replay to be staged, citing a decision to invalidate the result of a World Cup qualifier between Uzbekistan and Bahrain in 2005 on the basis of a "technical error by the referee".
"The Football Association of Ireland is hoping FIFA and its disciplinary committee will, on behalf of football fans worldwide, act in a similar fashion so the standards of fair play and integrity can be protected," added the FAI.
Earlier, Irish justice minister Dermot Ahern led calls for a replay, saying FIFA owed it to thousands of devastated fans around the country.
French Prime Minister Francois Fillon said it was not up to either government to get involved in FIFA decisions.
"France will obviously respect what the International Federation will decide," Fillon told reporters.
FIFA confirmed it had received the FAI's request and a spokesman told Reuters: "FIFA is not in a position to comment on decisions taken by match officials."
