Allardyce alleged that referee Phil Dowd favoured the hosts as he failed to award the Hammers a penalty in the third-round replay, before handing one to United towards the end of the match.
He told reporters following the match: "The one difference between Rafael's handball and Spence's was that Jordan plays for West Ham away against Manchester United and Rafael plays at home at Old Trafford.
"It's got to be that simple. Phil Dowd was in the perfect position. Rafael pushes the ball away with his left arm. No penalty. Go to the other end and the ball hits Jordan's hand. If you give one you have to give both, simple as that."
As a result of his words, the FA have fined the Englishman £8,000.