Cabaye had denied deliberately kicking out at Albion defender Adam El-Abd at the Amex Stadium and contested the FA charge.
The Brighton player said: "I don't believe he kicked out to hurt me, it seemed an accident."
But a regulatory commission found him guilty and he missed Wednesday night's Barclays Premier League game at Blackburn and the upcoming matches against Aston Villa and Tottenham.
Cabaye tweeted: "3 matchs ban it's a real injustice but life goes on!!! In my teammates i trust ..."
Alan Pardew: "I had a call from the Brighton coaching team to suggest to me that their player wants to support our player, and I have never had that.
"We never instigated it, we never asked or put a phone call in to Brighton.
"We have sent our letter in, a strong letter, and we have also, after the contact from Brighton, asked the Brighton player if he would forward a letter to us, which he has kindly done.
"If we had 18 cameras at every game, you would be sending two or three players off at every set play because there's contract, trust me, more intended contact than there is with Cabaye's incident.
"I really do think they need to just take a deep breath on this one, have a good look at it."