Kompany, with FA Cup holders City already 1-0 down, was sent off in the 12th minute at Eastlands after going in two-footed to take possession off Nani.
Although he won the ball cleanly, without appearing to make contact with Nani, the defender was shown a straight red card by referee Chris Foy for a two-footed challenge.
As the Ciy skipper has already been sent off for a straight red card offence once this season, he is facing a four-match ban unless he can get the red card card rescinded.
"We will appeal," said Mancini. "I am sure we will win."
This third round loss meant City's defence of the FA Cup lasted just one match but the Premier League leaders remain three points clear of second-placed defending champions United at the top of the table.