The punishment was served to the Spain international retrospectively, after the FA reviewed the incidents that happened before half time at Stamford Bridge on Saturday.
At the time, Costa appealed to referee Mike Dean, that Arsenal defender Gabriel Paulista, had kicked him and it led to the sending off of the Gunners player.
The FA has now banned him for striking Laurent Koscielny in the face, which means he will miss Chelsea’s Capital One Cup trip to Walsall and their matches against Newcastle and Southampton.
It is the second time Costa has been suspended via retrospective action and the club have said they are “extremely disappointed” with the verdict.
The Chelsea statement read: “We are extremely disappointed with the FA Regulatory Commission’s decision to suspend Diego Costa. We will await their written reasons before commenting further.”