The defender would have faced a three-game suspension, missing matches against Fulham and Queens Park Rangers in the Premier League, and an FA Cup fourth round fixture against Crystal Palace or Stoke.
The 26-year-old was dismissed by referee Mike Dean in the second half of the 2-0 win, but defended his challenge on Twitter after the game: "If the ball is overrun by the opponent and a 50/50 challenge occurs, [a] collision is inevitable."
A FA statement read: "An Independent Regulatory Commission has upheld a claim of wrongful dismissal regarding Manchester City's Vincent Kompany.
"The defender was dismissed for serious foul play during his side's match against Arsenal on Sunday 13 January 2013.
"As a result, Kompany's three-match suspension has been withdrawn with immediate effect."