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"Having the red card rescinded goes some way to rectifying the error and we are delighted to have Wes available for Saturday's game at Aston Villa."
The FA also revealed that Crystal Palace were unsuccessful with their appeal against the red card Eagles midfielder Yannick Bolasie received in their top-flight game at Hull on Saturday.
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.
A statement from the FA said: "An Independent Regulatory Commission has heard two separate claims of wrongful dismissal involving Sunderland's Wes Brown and Yannick Bolasie of Crystal Palace.
"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 As a result, his three-match suspension has been withdrawn with immediate effect.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."
PREMIER LEAGUE FIXTURES
Saturday November 30
15:00 Aston Villa v Sunderland
15:00 Cardiff City v Arsenal
15:00 Everton v Stoke City
15:00 Norwich City v Crystal Palace
15:00 West Ham United v Fulham
17:30 Newcastle United v West Brom
Sunday December 1
12:00 Tottenham v Man Utd
14:05 Hull City v Liverpool
16:10 Chelsea v Southampton
16:10 Manchester City v Swansea City
Tuesday December 3
20:00 Crystal Palace v West Ham United
Wednesday December 4
19:45 Arsenal v Hull City
19:45 Liverpool v Norwich City
19:45 Man Utd v Everton
19:45 Southampton v Aston Villa
19:45 Stoke City v Cardiff City
19:45 Sunderland v Chelsea
19:45 Swansea City v Newcastle United
20:00 Fulham v Tottenham
20:00 West Brom v Manchester City
Sunderland Fixtures
Saturday November 30
15:00 Premier League Aston Villa v Sunderland
Wednesday December 4
19:45 Premier League Sunderland v Chelsea
Saturday December 7
17:30 Premier League Sunderland v Tottenham
Saturday December 14
15:00 Premier League West Ham United v Sunderland
Tuesday December 17
19:45 League Cup - Qtr Sunderland v Chelsea
Saturday December 21
15:00 Premier League Sunderland v Norwich City
Thursday December 26
15:00 Premier League Everton v Sunderland
Saturday December 28
17:30 Premier League Cardiff City v Sunderland