Mousa Dembélé will serve a six-match suspension with immediate effect after he accepted a charge of violent conduct.
The Tottenham Hotspur midfielder had appeared to claw his fingers down the face of Diego Costa, the Chelsea striker, during the tempestuous 2-2 draw at Stamford Bridge on Monday night.
The incident was not seen by the match officials but was caught on video.
Dembélé did not appeal but asked for a personal hearing after which the FA deemed that an additional three-match suspension was appropriate.
The decision means that the Belgian will now be unavailable to Spurs until their fifth game of next season.
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Mauricio Pochettino has admitted that Tottenham crossed the line in a number of brawls with Chelsea on Monday but said that it proves his team can no longer be dismissed as soft.
Both Tottenham and Chelsea were charged with failing to control their players while nine Tottenham players were booked, a record for a single team in a Barclays Premier League match.
“Sometimes we cross the line and if you cross the line you need to pay,” Pochettino said. “It’s a good lesson for us that we need to learn for next season and I’m sure we will be more strong next season. People who play football know this situation can happen.
“Maybe we agree in theory but it’s difficult to apply when you contest the title. It’s difficult to keep calm. I am proud of my players. We show character. Some people tried to put calm to the situation, some behaved more aggressive.”
The incident comes in the wake of Dele Alli being found guilty of violent conduct last week after he punched Claudio Yacob, of West Bromwich Albion, and will now miss the remainder of the season.
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“Sometimes when you run a lot and you’re under stress, you can cross the line,” he said. “If you cross the line Mousa knows you need to pay, like Dele Alli. This happens in football. We cannot justify it but we can understand it.”
Pochettino believes that the Chelsea clash was the latest example of an opponent trying to provoke Tottenham, which he takes as a compliment and feels that the club has changed its image as a team who roll over easily.
“We showed character and quality enough to be a contender for the Premier League,” he added. “But if you want to be a champion and want to improve and be a real contender for the league you know that something [like this can] happen.
“I’ve heard over the last few years that in the past Tottenham were soft, but now we start to show character. We cross the line maybe, yes, I’m not stupid but it’s important to manage our energy better for the next season and be sure that we learn a lot.
“This season we showed that we are not a nice team to play. Not anymore. Our opponents don’t like to play against us because we showed passion, maybe too much passion. But we need to take it in a positive way.
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“This season we created a very good feeling and energy between our supporters and staff and players. We need to keep this and we need to manage it in a better way and try to win the title.”