The broadcaster, whose distinctive Ulster accent made him a regular subject for satirists, spent more than 10 years with the BBC after a lengthy career in newspapers.
His family said: “While many people will remember John for his journalism and broadcasting, for us he was the most loving, funny and devoted husband, father and grandfather. We will miss him terribly, but have so many memories of the tremendous happiness he has brought into our lives.”
In his memoirs, Cole recounted how an interview early in his career with then prime minister Clement Attlee during a holiday in Ireland inspired him to to work in Westminster.
He went on to interview many of Westminster’s biggest names and was a regular fixture outside Downing Street in the 1980s.
The BBC’s current political editor Nick Robinson said Cole’s death was “sad news”.
Writing on Twitter, he said: “The man I learnt so much from, the BBC’s former political editor John Cole, has died. He shaped the way all in my trade do our jobs.”
Scotland First Minister Alex Salmond said: “When I started out in politics at the House of Commons, John Cole was the main BBC correspondent. Not only was he an extremely able journalist but also extraordinarily helpful and generous to a young politician.”
Labour leader Ed Miliband wrote on Twitter: “My generation grew up watching John Cole. He conveyed the drama and importance of politics. Many condolences to his family.”