He later became a popular television commentator in Australia.
"He was a massive character," said BBC cricket correspondent Jonathan Agnew. "Whatever he did, Tony was huge - as a character, as a man, as a cricketer."
Former England captain Sir Ian Botham described Greig as "an amazing guy, just full of energy".
"He revolutionised the game and it had to be done," he added.
Australian broadcaster Channel Nine said Greig died at about 13:45 local time on Saturday, after being rushed from his home to St Vincent's Hospital earlier in the day.
Greig, who stood 6ft 6in, scored 3,599 Test runs at an average of 40.43, took 141 wickets with his off-spin and medium pace at 32.20 apiece, and was named one of Wisden's Cricketers of the Year in 1975.
Tony Greig's career
England, Border, Eastern Province, Sussex
3,599 (average 40.43)
141 (average 32.20)
(average 31.19) and took
(average 28.85) in 350 first-class matches
He told Channel Nine colleagues before having surgery last month: "It's not good. The truth is I've got lung cancer. Now it's a case of what they can do."
He tweeted on Christmas Day:
"Merry Christmas and a Happy, Healthy and Prosperous New Year to you all. Would love to be at Test but son Tom and I will be tuned in."
Greig, who played for Sussex and qualified for England through his Scottish parents, provoked controversy in 1976 when, as England captain, he said he intended to make West Indies "grovel" in the home Test series.
England were beaten 3-0 but the following winter he led them to their first series victory in India since the Second World War.
After presiding over three wins, six draws and five defeats he relinquished the England captaincy in 1977.
Greig was a central figure in recruiting several England players for the
controversial World Series,
which ran in opposition to Test cricket from 1977-79 and featured international stars earning much higher salaries.
Although several players were banned from representing their country, World Series Cricket helped revolutionise the sport with increased player wages and presentational changes such as the introduction of coloured clothing.
World Series Cricket
The first year of World Series Cricket in 1977 saw two series of five-day matches, which were not allowed to be called Tests and had to be termed "SuperTests", Australia losing them both to West Indies and a World XI. The three teams also took part in a round-robin one-day tournament, won by West Indies. The following year the format changed to four-day matches played as day-night games, the World XI winning a round-robin format and West Indies retaining the one-day title.
Greig lived in Sydney from the late 1970s until his death and became a popular voice around the world with his enthusiastic and opinionated commentary style for Channel Nine, often wearing a large panama hat and inserting car keys into playing surfaces as part of his pitch reports.
"He changed cricket in the way we know it now," said fellow broadcaster Botham. "The players of today have a lot to thank Tony Greig - and Kerry Packer - for.
"Flamboyant is the word - he was larger than life and very much an extrovert. He made things happen."
England & Wales Cricket Board chairman Giles Clarke said: "Tony Greig was a magnificent and fearless cricketer capable of changing games with ball or bat. He was a determined supporter of players' rights in his later years."
Former Australia captain Bill Lawry, a long-term colleague in the commentary box, said: "He's been a great friend of mine for 33 years.
"He's well known right throughout the world, well loved and respected. World cricket has lost one of its great ambassadors."
1975 Ashes archive - Lillee and Thomson star for Australia as Greig becomes captain
Australia captain Michael Clarke said: "I was only speaking with Tony a couple of days ago so news of his passing is absolutely devastating. He has been a great mentor for me.
"Cricket will be much poorer for his loss. We will never forget the lasting legacy Tony leaves us with."
Australia Prime Minister Julia Gillard described Greig as "a wonderful example of someone who came to Australia from somewhere else in the world and embraced his adopted country as his own".
International Cricket Council chief executive Dave Richardson said: "Tony played a significant part in shaping modern cricket as a player in the 1970s and then provided millions of cricket lovers with a unique insight as a thoughtful and knowledgeable commentator."
Current England Test and Sussex wicketkeeper Matt Prior
"Can't believe one of my heroes Tony Greig has passed away. One of the greatest voices in cricket and will be sorely missed. #RIPGreigy."
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