Brown was shown a red card for his tackle on Charlie Adam at the Britannia Stadium, a decision that enraged manager Gus Poyet.
However, the FA confirmed in a statement that Brown's appeal had been upheld, adding that the three-match suspension had been "withdrawn with immediate effect".
Crystal Palace had also appealed a ban handed out to winger Yannick Bolasie, after he was dismissed for a clumsy challenge on Hull City's Jake Livermore.
However, the winger's appeal was dismissed and he will sit out the next three games.
Poyet urged Mike Riley, general manager of the Professional Game Match Officials Limited, to contact him about the decision, after it was revealed Riley had done the same following West Brom's Steven Reid's challenge against Ramires at Stamford Bridge.
The Uruguayan confirmed Riley had spoken to him, adding that he was pleased to welcome back Brown for the clash with Aston Villa on Saturday.
"I would like to express my thanks to the panel for giving the matter their consideration. Naturally we are delighted with their decision, which is fair and just," he told the club's official website.
"I have spoken with Mike Riley and whilst the details of our conversation will remain private, I am satisfied with today’s outcome.
"Of course it was hugely frustrating to have Wes sent off, but we accept that officials are only human and like everyone, they can make mistakes.
"Having the red card rescinded today goes some way to rectifying the error and we are delighted to have Wes available for Saturday’s game at Aston Villa."
