The defender clattered into Jack Wilshere with feet raised, although the sliding challenge did win the ball.
"We will appeal, because it is impossible that we can lose one player for three games for nothing. It is not red card, it is nothing," said City boss Roberto Mancini.
"He went into the tackle, anticipating the opponent and took the ball before.
"I don't know how it is possible to get a red card like this. It was no foul, no red card.
"He went in with one foot, that is a tackle. If you want to be right and just, we can say what has happened - if it was a red card, I would say, but it was not a red card, because he anticipated Wilshere. Totally, 100%."
Kompany took to Twitter to give his side of the story, accompanied by a photo which appeared to show one of his feet being on the floor as he made the tackle.
"No grudges against the referee, I understand the difficulty of the job," he said.
City remain seven points behind United, and Mancini insists all is not yet lost.
"At this moment, United play very well, but the season is long. It is important for us to be there, close with them, and we will arrive the moment they lose points," he said. "We have the derby on April 6 and we need to arrive there with five or four points behind, it is okay."