The FA announced on Tuesday that the Wearside outfit's appeal against their defender's sending-off had been upheld, meaning he is now free to be selected for the Barclays Premier League game at Aston Villa on Saturday.
And giving his reaction, Poyet said on Sunderland's official website: "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."
Brown was dismissed in the first half of Saturday's top-flight contest at the Britannia Stadium, which Sunderland lost 2-0, for a sliding challenge on Charlie Adam.
The Black Cats man, travelling at speed when he made the one-footed tackle, won the ball and replays suggested any contact between the two players had been minimal.
Referee Kevin Friend deemed it worthy of a straight red card - a decision that left Poyet livid and calling for an apology from Mike Riley, the head of the Professional Game Match Officials Board.
Friend has not been given a match for the next two rounds of Premier League games.
Poyet added on Tuesday: "I have spoken with Mike Riley and, whilst the details of our conversation will remain private, I am satisfied with the 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"Having the red card rescinded goes some way to rectifying the error and we are delighted to have Wes available for Saturday's game at Aston Villa."
Bolasie will now serve a three-match suspension.
His straight red card was issued by referee Anthony Taylor for a challenge on Jake Livermore 12 minutes from the end of the match at the KC Stadium, which Palace won 1-0.
"With regard to Brown, who was dismissed for serious foul play during Saturday's game at Stoke City, the claim was upheld. As a result, his three-match suspension has been withdrawn with immediate effect.
"The claim in relation to Bolasie, who was dismissed for serious foul play during his side's match at Hull City on Saturday, was unsuccessful therefore his three-match ban remains with immediate effect."