The decision to rule that referee Mike Dean had made an error was the latest in a rash of judgements this season which has seen the FA's regulatory commission overturn no fewer than six red cards for top-flight players. Kompany was sent off for a two-footed tackle on Arsenal's Jack Wilshere and the FA's decision has been privately condemned by some officials.
Elleray, chairman of the FA's referees committee, said: "When you have a referee's decision overturned then some people will be disappointed but ultimately there's a process which has to be gone through. We always say to referees to do their job on the field of play and not to worry about anything that happens afterwards."
He added: "Football's history is full of disciplinary decisions that have been overturned but referees are not being influenced by decisions by the regulatory commissions.
"I think there is disappointment in the decision by referees but they accept there is an off-the-field process and there are also certain occasions where clubs, players and managers are disappointed when red cards are supported."
Elleray said some decisions were judgement calls which were extremely difficult to make in a split second but that is "the beauty" of the game.
He added: "The issue is that there will always be some incidents in football which are not black and white. And if you step back that's the beauty of football because people then have things to discuss.
"You have to remember the referee makes an honest decision in real time and they won't get the decision right all the time. If you look at the second Manchester United goal on Sunday, if you slow it down there's a case for offside but it would be impossible for anyone to see that in real time. Referees make judgements in a split second."
Apart from Kompany, the other players to have red cards overturned this season were Carlton Cole and Darron Gibson in last month's game between West Ham and Everton, Stoke's Steven Nzonzi, Jordi Gomez of Wigan and Tom Huddlestone of Tottenham.