Grealish, who won 45 Ireland caps during a 10-year international career, had been suffering from cancer.
The highlight of the London-born midfielder's career came in 1983, when he captained Brighton to the FA Cup final against Manchester United.
And a statement on the Championship club's website read: "Brighton & Hove Albion are saddened by the news that former captain Tony Grealish has passed away.
"On behalf of everyone at the club, our thoughts go out to partner Wendy, children Garret and Jordana, grandchildren Harrison and Joshua, likewise Tony's mother Nora, brother Brian and sisters Anne and Christine at this very sad time."
Grealish, whose career began at Leyton Orient in 1974, also played for Luton, Manchester City, Rotherham and Walsall.