John McCafferty was told he had only five years to live when he received the transplant at Harefield Hospital in west London, on 20 October 1982.
His widow Ann said: "The last 30 years we had together were brilliant. We've travelled the world."
Mr McCafferty, from Newport Pagnell in Buckinghamshire, died aged 73 on Tuesday at Milton Keynes Hospital.
Live: Heart transplant patient wanted to be an 'inspiration'
He was officially recognised as the world's longest surviving heart transplant patient by Guinness World Records in 2013.
At the time he said: "I want this world record to be an inspiration to anyone awaiting a heart transplant and to those who, like me, have been fortunate enough to have had one."
Mr McCafferty was diagnosed with dilated cardiomyopathy when he was 39.
The condition leads to scarring of the heart wall and damage to the muscle which causes the heart to become weakened and enlarged, preventing it from pumping efficiently.
The transplant was carried out by surgeon Sir Magdi Yacoub.
In the years that followed, Mrs McCafferty said her husband had taken part in the British Transplant Games and "did so much".
The couple celebrated their 50th wedding anniversary in October.
The first-ever successful heart transplant operation was performed in South Africa in 1967 by Prof Christiaan Neethling Barnard and a team of 30 physicians at the Groote Schuur Hospital, Cape Town.
The patient, Louis Washkansky, survived for 18 days with the new heart.
The world's longest-surviving heart transplant patient has died, 33 years after his life-saving operation.