Harrington, who made over 400 appearances for Cardiff during a one-club career between 1951 and 1966, is regarded as one of the Bluebirds' greatest-ever players.
Penarth-born Harrington won 11 caps for Wales and helped Jimmy Murphy's side qualify for the 1958 World Cup - the country's first major tournament.
Harrington played in both World Cup play-off games as Wales beat Israel in a two-legged tie, but he missed the finals in Sweden the following summer after dislocating his shoulder.
"The FAW is saddened to learn of the passing of Alan Harrington," the Football Association of Wales posted on their official Twitter account.
"Our thoughts are with Alan's family and friends at this time."
Cardiff paid their own tribute, saying: "The club has been saddened by the news of the passing of one of our all time greats, Alan Harrington, who spent his entire club career with the #Bluebirds between 1951-66.
"Our sympathies go to Alan's family at this very difficult time."
Harrington managed Barry in the Welsh League after his playing career and worked as a purchasing manager for a chemical company in Cardiff.
He is survived by his wife Gloria and daughters Judith and Rachel.