Costa denied the charge, relating to an incident involving Laurent Koscielny during Saturday’s 2-0 win over Arsenal.
Costa put his hands in Koscielny’s face before clashing with Gabriel, who was sent off but had a three-game ban overturned earlier on Tuesday.
In a statement, Chelsea said they were ‘disappointed’ with the verdict.
Costa’s ban covers Chelsea’s League Cup match at Walsall on Wednesday along with Premier League matches against Newcastle and Southampton.
Having confirmed the match officials did not see the incident between Costa and Koscielny, the FA supplied video footage to a panel of three former elite referees.
Each had to independently rule Costa’s actions to be an instant red card for the FA to proceed with the charge.
The FA has charged Gabriel with improper conduct for failing to leave the pitch immediately after being sent off, and both clubs with failing to control their players.
