The Scottish actor was best known for his portrayal of James Bond, being the first to bring the role to the big screen and appearing in seven of the spy thrillers.
Sir Sean died peacefully in his sleep in the Bahamas, having been “unwell for some time”, his son said.
His acting career spanned five decades and he won an Oscar in 1988 for his role in The Untouchables.
Sir Sean’s other films included The Hunt for Red October, Highlander, Indiana Jones and the Last Crusade and The Rock.
Jason Connery said his father “had many of his family who could be in the Bahamas around him” when he died overnight in Nassau. Much of Bond film Thunderball had been filmed there.
He said: “We are all working at understanding this huge event as it only happened so recently, even though my dad has been unwell for some time.
“A sad day for all who knew and loved my dad and a sad loss for all people around the world who enjoyed the wonderful gift he had as an actor.”
His publicist Nancy Seltzer said: “There will be a private ceremony followed by a memorial yet to be planned once the virus has ended.”
He leaves wife Micheline and sons Jason and Stephane.