He had been in a critical condition with an intestinal infection since being admitted to intensive care on Friday at a hospital in Sao Paulo.
Socrates, who was widely regarded as one of the greatest ever midfielders, was moved onto a life support machine on Saturday.
He played in two World Cups, won 60 caps for his country between 1979 and 1986 and scored 22 goals.
The former Corinthians player, whose full name was Socrates Brasileiro Sampaio de Sousa Vieira de Oliveira, was taken to the Albert Einstein Hospital in Sao Paulo with food poisoning on Friday, according to his wife.
A hospital statement said on Saturday that the former footballer was "in a critical condition due to a septic shock of intestinal origin".
It added he was breathing with a ventilator and using a dialysis machine.
Brazilian president Dilma Rousseff said Brazil had lost "one of its most cherished sons".
"On the field, with his talent and sophisticated touches, he was a genius," she said. "Off the field... he was active politically, concerned with his people and his country."
Socrates scored 172 goals in 297 games for Brazilian club side Corinthians, having begun his career at Botafogo.
More than a decade after retiring, he joined non-League Garforth Town at the age of 50 on a one-month deal as player-coach, but managed just 12 minutes as a substitute.
Zico, a team-mate of Socrates in the iconic Brazil side of the 1970s, told website globoesporte.com: "He was a spectacular guy.
"As a player, there is not much to say; he was one of the best that I ever played with. His intelligence was unique."
Paolo Rossi, who scored a hat-trick in Italy's memorable 3-2 quarter-final win over Socrates's Brazil in 1982 - widely regarded as one of the greatest games in World Cup history - paid tribute to his former opponent.
"It's a piece of our history that's broken off and gone away," he said.
"Socrates seemed like a player from another era. You couldn't place him in any category - on the pitch and even more so off it.
"Everyone knew about his degree in medicine and he had a lot of cultural and social interests as well. He was unique from every point of view."
Socrates scored Brazil's first goal in that match, beating Dino Zoff at the near post after running on to Zico's wonderful through ball.
"I remember the goal he scored against Zoff; he was one on one and it didn't seem like he could get to the ball. He looked slow but in reality he wasn't.
"He was a very dynamic player with a sublime foot but, most of all, great intelligence."
Fiorentina, where Socrates spent one season, held a minute's silence before the Serie A match against Roma, when the players wore black armbands.
A club statement read: "To the unforgettable 'Doctor' who played with the purple shirt in 1984/85, playing 25 matches and scoring six goals and who will always be remembered for his footballing intelligence, he will be affectionately remembered by the club, the team and the Fiorentina fans."
Giancarlo Antognoni, a team-mate of Socrates at Fiorentina and an opponent in that famous 1982 World Cup clash, said: "I'm really hurt.
"He was a true personality, above the rules with his own methodology, his way of life and his ideas.
"He struggled to adapt to our football but he was an authentic champion, full of refined class, great charisma and character."
Giancarlo De Sista, Fiorentina's coach at the time, also spoke of his admiration for Socrates as a player and a person.
"Socrates was a very intelligent man; he had great class," he said.
"I remember that he was an objector. He wanted to know everything - why he couldn't smoke on the team bus, why we had to be in retreat on the Saturday nights before games.
"He was an intelligent person who was interested in politics, although he smoked and drank a bit too much."
Former Brazil striker Ronaldo wrote on Twitter: "Sad start to the day. Rest in peace Dr. Socrates."
Socrates was taken to hospital in August and September this year with bleeding in his digestive tract.
After these incidents he admitted he had problems with alcohol, especially so during his playing career. He is also well known for his smoking habit.
In a recent television interview, Socrates said he had considered alcohol his "companion" but believed its regular use did not affect his performance on the field.
"Alcohol did not affect my career, in part because I never had the physical build to play this game," he said.
"Soccer became my profession only when I was already 24. I was too thin and when I was young I did not have the opportunity to prepare myself physically for the sport."
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