Costa, 26, put his hands in the face of Arsenal centre-back Laurent Koscielny prior to clashing with Gunners' defender Gabriel, who was sent off.
The FA has also charged Gabriel with improper conduct and both clubs with failing to control their players.
Arsenal are to appeal Gabriel's dismissal and three-match ban.
Costa has until 18:00 BST on Tuesday, 22 September to respond to the charge, which is for the incident with Koscielny as it was missed by match officials but caught on video.
"Off the ball incidents which are not seen at the time by the match officials are referred to a panel of three former elite referees," said an FA statement.
"Each referee panel member will review the video footage independently of one another to determine whether they consider it a sending-off offence.
"For retrospective action to be taken, and an FA charge to follow, the decision by the panel must be unanimous."
Referee Mike Dean booked both Costa and Gabriel for their spat, before the latter was dismissed for kicking out at the Blues forward.
Gabriel and both clubs have until 18:00 BST on Thursday, 24 September to reply to their charges.
Arsenal midfielder Santi Cazorla was sent off in the match for tw
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