The night of New Year’s Eve, 1958, began promisingly for the 33-year-old Magnum photographer Burt Glinn, who was at a black-tie party thrown by a New York Times reporter on Manhattan’s Upper West Side. As the drink flowed, talk focused on the US-supported Cuban dictator, Fulgencio Batista. Word had it that he was on his way into exile. Glinn, who died in April this year, knew that Fidel Castro, the leader of the revolutionaries, was up in the mountains of the Sierra Maestra and would be reachable. Without a second thought, he was off: borrowing money, he caught the last shuttle to Miami and then begged his way on to a flight to Havana. He touched down at dawn.

“By the time I arrived,” he later wrote, “Batista had fled. Fidel was still hundreds of miles away, although nobody knew exactly where. Che Guevara was on his way to Havana and nobody seemed to be in charge… There was a lot of firing. You could not tell who was shooting at whom. People appeared in the streets wearing “26 Julio” armbands [the name of Fidel’s movement]… Castro called a general strike to show the old guard who was really in charge. This may have been great for the revolution, but it was bad news for the journalists. For the next five days, all the stores and restaurants were closed… Fortunately in Havana, no matter what, you can always find a cigar and a bottle of rum. That is primarily what kept us going.”

Glinn teamed up with French photographer Bob Henriques (who later became a Magnum associate), and they managed to buy a clapped-out taxi for $100 in which they set off for the hills in search of Castro. The remarkable photographs Glinn and Henriques took during that trip are among the classics of the Cuban revolution, encapsulating the exhilaration and wild hopes of that extraordinary time. We see Castro waving from his Jeep at the thronging crowds of Cuban civilians, soldiers embracing, and a caravan of vehicles on the long journey into Havana. Along with work from Cuba by other Magnum photographers, including René Burri, Andrew Saint-George and Elliott Erwitt, the photographs will be on display in London at an exhibition of vintage and contemporary prints marking the 50th anniversary of the Cuban revolution.

One of Glinn’s finest photographs shows a young guerrillero who had come down into Havana having been hiding for months up in the hills. He is photographed flirting, rifle slung casually over his shoulder, with a young woman dressed in a tightly fitting dress and heels. “Burt loved that picture,” recalls his widow, Elena Glinn. “He used to say, ‘I don’t know what she looks like now, but she looked pretty good then.’ I also remember him saying Fidel always managed to surround himself with the most attractive women soldiers.”

Only three Western photographers joined Castro’s triumphant march towards Havana; Glinn was one of them. As Castro neared the capital, crowds swelled. “At Santa Clara, about 180 miles from Havana, the momentum really grew,” wrote Glinn. “The column arrived in a disorganised frenzy. Nobody knew which car Castro was in, but they were delirious. Santa Clara had been the site of the one big battle of the revolution, and when Che Guevara took the city, it was a signal to the Army that it was finished, and Batista fled. Although the city was already in rebel hands, Castro’s arrival was greeted like the liberation of Paris… By the time he entered Havana, the crush was so great that I lost my shoes while struggling to get my pictures.”

René Burri, another celebrated Magnum photographer, got into Cuba early in 1963, at the peak of the Cold War confrontation, when a New York reporter was granted an interview with Che Guevara, and Burri was sent to photograph him. It was another New Year’s Eve flit. “I was about to sit down for a dinner in Zurich when I got a call telling me I had to get on a plane to Cuba,” he recalls. “Guevara was Minister for Economics, Fidel’s right-hand man. His portrait was on the banknotes. He and the reporter became locked in discussion for two hours. It was very aggressive – like a cock fight. Guevara stomped around in his office like a caged tiger, and suddenly in the middle of it, he turned and said to me, ‘If I catch your friend Andrew [Saint-George, who was suspected of being a CIA agent], I’ll cut his throat.’”

Burri’s portrait has been reproduced widely in the print media, but also featured on T-shirts, mugs and elsewhere. “I went back to Havana earlier this year and gave a signed copy of the portrait to Castro. He started growling at it. ‘What’s the matter?’ I asked him. ‘Just look at me, and just look at you,’ he said. ‘We have grey hair. And look at him…’ It was pure Fidel.” Guevara was captured with the help of the CIA and executed in Bolivia in 1967. He was, in many ways, the idol of the revolution, and his face, well known from Burri’s portrait, became a wider symbol of rebellion.

Elliott Erwitt went to Cuba in 1964 and spent a week with Castro alongside a Canadian film crew. “He was like a benevolent cowboy. He did lots of mixing with the people, pitching balls in baseball games and kissing babies. It all seemed quite normal. Havana was full of broken-down cars and peeling paint, just as much of a falling-apart town as it is today, but there was a very good spirit about the place then.”

“Everyone was so enthusiastic,” adds Burri. “It’s ironic because so soon afterwards things started to go badly. The worst time was the Seventies, when Fidel had no option but to become friends with the Russians. I remember standing near him when he was seeing [Leonid] Brezhnev off at Havana airport. Fidel waved, the aircraft taxied away and took off, and then Fidel went into a kind of tantrum on the tarmac, stamping, growling and swearing. He was so completely in the hands of the Soviets, and there was nothing he could do.”

Cuba: 50 Years of Revolution is at the Magnum Print Room, 63 Gee Street, London EC1, from December 2-January 31, 2009 (020-7490 1771; www.magnumphotos.com/exhibitions).

René Burri talks about his work in Cuba at the Cochrane Theatre, London WC1, on December 4 (www.cochranetheatre.co.uk)

Editor's choice

• The super cool Vampire Weekend
• Streep and Baldwin get complicated
• Guy Ritchie on Sherlock Holmes