Allardyce was angry after West Ham were denied what he considered a penalty and United were awarded a spot kick, a decision he put down to Alex Ferguson's side enjoying home advantage in the third-round replay.
"Allardyce was £8,000 and warned as to his future conduct," the FA, which had earlier charged the manager with improper conduct over comments made to the media, said on its website.
West Ham were not given a penalty in the all-Premier League tie when the ball struck United defender Rafael's arm but the hosts got one when Jordan Spence handled late. Wayne Rooney missed the spot-kick.
"You see it time and time again at Old Trafford," Allardyce said after the game.
"There's no doubt about the difference between Rafael's handball and Jordan Spence's. Spence plays for West Ham and the away team, while Rafael plays for the home side at Old Trafford.
"He (Dowd) couldn't find the courage to give it unfortunately, and there was a very similar incident with Jordan Spence and it was given. You have to have the courage to referee here. But why are you a professional referee or player? To come to these places to do your job and do it to the best of your ability."
Allardyce had denied the misconduct charge and requested a personal hearing.