In February 1968, the avant-garde legend Marcel Duchamp hosted an informal dinner in New York for a group of local artists. Among them were composer John Cage, choreographer Merce Cunningham and the painter Jasper Johns – who, that night, was working up the courage to ask Duchamp a favour. He wanted to create a set design for a Cunningham dance that would use images from Duchamp's 1923 work The Large Glass (also known as The Bride Stripped Bare by Her Bachelors, Even).

All those present, apart from Johns, have since died. But a few years ago, when I asked Cunningham about it, he remembered the moment precisely: "John was playing chess with Marcel's wife Teeny, and Marcel was smoking a cigar, watching. Jasper went over to Marcel and asked if he could make the set, and Marcel said, 'Yes, but who would do all the work?' Jasper said, 'I would.'"

Cunningham's description was a typically understated account of a scene that would have awed many. By 1968, he, Cage and Johns had caused a revolution in the art of performance. Working in the Dadaist traditions Duchamp had helped to create, their collaborations had combined dance, music and design in groundbreaking ways. Choreographers like Yvonne Rainer and Lucinda Childs were among the legions of artists influenced by them. As Childs has said: "They were giants to us."

Johns's Duchamp-inspired set featured in Walkaround Time, a dance of strange vibrating beauty in which seven inflatable vinyl rectangles were imprinted with fantastical mechanistic imagery drawn from The Large Glass. These seemed to float, magically, in the vinyl. Yet the inflatables themselves, which were strewn around the stage, were so large that, far from being a frame for the choreography, they became like a maze the dancers had to navigate. Nothing like it had ever been seen. And the continuing resonance of this work is one of many elements in Dancing Around the Bride, an exhibition celebrating the cross-currents of influence between Duchamp and these New Yorkers. Currently showing at the Philadelphia Museum of Art, it transfers to the Barbican next month.

One strand dates back to 1942, when Cunningham and his new lover Cage began working together, experimenting with mixed-media works that played on the boundaries between art and real life. The two men gleefully incorporated everyday objects, sounds and movements, and introduced elements of chance – throwing coins or consulting the I Ching – to shape their compositions. In among the many artworks and manuscripts on show are the notes and diagrams Cunningham made for his 1956 piece Suite for Five, vividly depicting the storm of ideas and impulses that went into its creation.

In the early 1950s, the two men became close to Johns and his then lover, the painter Robert Rauschenberg. This quartet would work closely together for the next four decades, united by the dance company Cunningham formed in 1953; part of what makes this show such a fresh and clever record of their association is the fact that live performance plays a key role. A large central room houses a stage and, on the day I visit, I catch Daniel Squire, a former member of the company, performing a solo culled from extracts of Cunningham's repertory. As Squire twists, tilts and skitters through the fine detail, he demonstrates its extraordinary fusion of the classical and the unexpected – imagery that turns his body fleetingly from divine athlete to frantic pedestrian, or something like a long-legged wading bird. Nearby, an automatic Disklavier piano produces an oddly lyrical cacophony: a fragment of music by Cage with its own sweetly disordered logic.

Around the room are works by Johns and Rauschenberg; not only the set for Walkaround Time, but also the chairs-and-bicycle-wheels installation Rauschenberg created for Cunningham's 1977 piece Travelogue, as well as the wood-and-newspaper screens he made for 1954's Minutiae. Meanwhile, hanging on the wall, is Rauschenberg's Express, his 1963 collage that includes a photograph of Cunningham's company. Their presence makes Squire feel as if he's dancing in the presence of benign spirits: "It's lovely to feel Merce is here in this space, surrounded by his old friends."

It was Cage who first met Duchamp at a party given by Max Ernst and Peggy Guggenheim in 1942, though back then the French artist's work was known only to a select few. To others, he was better known as the professional chess player he became in the 1920s. In fact, Cage forged his own friendship with Duchamp by playing him regularly; they were both fascinated by the complex stratagems of chess, finding them a useful channel for communication.

Only slowly did the New Yorkers discover how many of their own experimental procedures had been pre-empted by Duchamp. As Cage wryly acknowledged in 1950: "Marcel was using chance operations the year I was born." In fact, as far back as 1913, Duchamp had composed music by writing individual notes on pieces of paper and pulling them out of a hat. More famously, he put a urinal in a gallery and called it Fountain; and in 1919, he exhibited a reproduction of the Mona Lisa with a moustache and goatee under the title L.H.O.O.Q. In French, this can be read as "Elle a chaud au cul", meaning "She has a hot ass".

These acts of sabotage and invention would inspire future generations. And one of the revelations of this exhibition is the degree to which Duchamp motifs were picked up by the four New Yorkers. The show's centerpiece is Duchamp's The Bride, a curious canvas painted in 1912 that imagined a woman in the form of a cubist machine. Its imagery reappeared in The Large Glass and it came to obsess Rauschenberg and Johns, the latter not only replicating it for Walkaround Time but returning to it in several other works.

Similarly, when Cunningham created his choreography for Walkaround Time, he included a playful "striptease" section that referenced Duchamp's 1912 cubist work Nude Descending a Staircase; the dance's serene tempi was also a nod to the artist. "Marcel always gave one the sense of a human being who is ever calm," Cunningham once said, "as though days could go by, minutes could go by."

The triumph of this exhibition lies in how it manages to capture the current of live creative energy that crackled between these men. Curator Carlos Basualdo took his inspiration from the four-hour dance "event" that was performed in New York shortly after Cunningham's death, in which his former dancers performed – and added to – fragments of his work. In London, his choreography will be performed by local dancers, including members of the Richard Alston Dance Company.

Basualdo also commissioned the French artist Philippe Parreno (co-creator of the acclaimed 2006 film Zidane) to keep the space animated, even when there are no dancers on site; this he does through sound and light. I catch one wonderfully evocative moment when the eerie notes of Cage's piano music are suddenly joined by the recorded sounds of dancing feet, just as an unseen mechanism flips open the window shutters, spilling daylight into the gallery along with a wave of street noise.

It's just the kind of serendipitous magic that these artists had been aiming for, long before their work became prized museum exhibits. And it puts me in mind of another story Cunningham told me, about a European revival of Minutiae. Rauschenberg's wood-and-newspaper set, by then priceless, could only be used on condition it was transported in a specially padded and air-conditioned van – a marked contrast to the 1950s, when the artists had simply strapped it to the roof of their old VW campervan. "We just hoped it wouldn't rain," said Cunningham.