Greig, who was battling lung cancer, died at St. Vincent's Hospital in Sydney, shortly after suffering the heart attack.
Hospital spokesman David Faktor told The Sunday Telegraph Greig's family was with him when he died.
Greig was born in South Africa and played in 58 Tests for England in the 1970s, including 14 as team captain.
His later role with the World Series cricket league, based in Australia, was credited with giving the staid sport a higher profile -- with higher player salaries and the introduction of colored clothing as opposed to all white uniforms.
"He revolutionized the game and it had to be done," Ian Botham, another former England skipper, told the BBC.
After his playing days wound down, Greig became a commentator on Australia's Channel Nine television.