On the Kutuzov embankment in St Petersburg is a handsome mansion that caught Mstislav Rostropovich's eye when he returned to Russia in 1990 after 16 years in exile. During the Soviet years it had been converted into communal apartments, and Rostropovich set about rehousing no fewer than 42 different families (the last of them moved out only last year) to allow him to restore the mansion to its former glory as a single residence. Now, four years after the cellist's death, his widow, Galina Vishnevskaya, still calls it home when she visits the city, and it continues to house their collections of art and furniture, as well as their archive of manuscript scores, newspaper cuttings, correspondence, photographs, medals and theatrical costumes. The only absentees are the cellos themselves, among them his "Duport" Stradivarius (reputedly once played by Napoleon) and his beloved Storioni, both safely stored in a vault.

On the top floor of the mansion, a curator has been painstakingly cataloguing a collection of video and audio tapes and discs, among them bootleg recordings of events that are sometimes hard to identify. When researching my new film Rostropovich: The Genius of the Cello, I was intrigued to find an almost silent VHS tape of him playing a concert, with Benjamin Britten conducting, shot from four different camera angles. The pictures also showed Shostakovich applauding effusively as both performers were wreathed in flowers. Frustratingly, the only sound on the tape was an electronic buzz.

The garlanded images of Britten and Rostropovich reminded me instantly of the photographs published after the premiere of Britten's Cello Symphony in Moscow. The Russian radio broadcast of that concert was issued some years ago on CD, but I didn't realise it might have been filmed until I saw that silent tape. So it was with a growing sense of excitement that I visited the Russian state film archives at Krasnogorsk, outside Moscow, to see what I could find. In Soviet days, the very existence of this archive was a state secret, but today the archivists are more accommodating. They dug out 11 minutes of footage from a Rostropovich-Britten concert, but without sound. Sure enough, it was the same footage as I had seen, but this time dated 12 March 1964, the day of the Cello Symphony's premiere. Four cameras had filmed the final three or four minutes of the work, but the rushes had never been edited together. Perhaps the sound fault had been too troublesome, the moment had passed, and the rolls of film put aside and forgotten.

Back in London, we put these images together, so we could watch all four cameras at once, and reunited them with the original radio sound. Even though the camera speeds were not uniform, it was as if Athena had sprung fully armed from the head of Zeus. We realised with a flood of emotion that we had recaptured a significant fragment of musical history, almost 50 years on. Here before our eyes we could witness, in Britten's own music, the extraordinary musical synergy between these two men, and the intensity of the occasion for both of them.

The Cello Symphony is not a work that is quick to dispense its favours, but the patient listener may well conclude that it is Britten's greatest orchestral work. As always with Britten, it took a particular performer to unlock his interest in the instrument. As a young man, and a viola player himself, Britten had never singled out the cello for special treatment, as he had Antonio Brosa's violin, Sylvia Spencer's oboe or Dennis Brain's horn. In America in 1941 he had contemplated writing a concerto for the great Austrian cellist Emanuel Feuermann, but nothing came of it, and it was not until he encountered the young Rostropovich at a 1960 concert in London that he fully grasped the cello's potential. It was effectively love at first sight.

Britten was already in the mood for love, primed by a radio broadcast he had listened to a few days beforehand. "This was a new way of playing the cello", he said, "almost a new, vital way of playing music." The concert itself featured the London premiere of Shostakovich's first cello concerto. From a fragment of news film shot by the BBC at the morning rehearsal, the sheer animal energy and abandon of Rostropovich at the age of 33 is fully evident. At the performance, Britten sat beside Shostakovich, who complained afterwards of bruised ribs, thanks to Britten repeatedly poking him with his elbow in delight at Rostropovich's playing.

How nearly it all went wrong. Rostropovich later confessed he had been barely aware of Britten, even by name. He had heard of his Purcell Variations, The Young Person's Guide to the Orchestra, but had clearly not actually heard the piece, since he thought Britten belonged to Purcell's era. So when Shostakovich offered to introduce him to Britten after the concert, Rostropovich thought it was a joke. He just managed to stop himself laughing as he turned round to face the notoriously sensitive British composer, and insisted (as was his wont with every composer he met) that Britten write him a cello work. Inspired by this "gloriously uninhibited musician", Britten agreed. The result was the Sonata for Cello and Piano, of which they gave the first performance 50 years ago – a new departure in Britten's music.

Although he self-deprecatingly warned Rostropovich of his lack of firsthand experience of the cello, Britten tests its technical boundaries in the Sonata with his use of harmonics, pizzicato and quadruple-stopping. He said Rostropovich freed him of his inhibitions, and certainly, after years of focus on the voice, there is an unbuttoned playfulness in his musical ideas. Yet at their first run-through in London, both musicians were nervous, and, according to Rostropovich, required "four or five very large whiskies" before they could begin. "We played like pigs," he said, "but we were so happy."

Britten went on to write three Suites for solo cello, in direct line of succession from the six of Bach: they are cornerstones of any cellist's repertoire. By now Rostropovich thought of Britten as a younger brother, even though Britten was 14 years his senior. Their shared love of music, cars, food and drink, let alone an almost puerile sense of the absurd, bound them together. Their friendship darkened and deepened with, first, Rostropovich's period of disgrace inside the Soviet Union and subsequent exile, and second, with Britten's heart operation and physical decline.

When Rostropovich himself was dying four years ago, his family played him some of his recordings, for stimulus and consolation. One was his performance with Britten of Schubert's Arpeggione Sonata, which had featured at their first recital together in July 1961. Ever since, any mention of Britten's name had brought a smile to his face, with the words: "Ah, Benjik!" Although Rostropovich was by now unconscious, his daughter Olga says that, as the music played, a tear slid down his cheek. More than the many other works they performed together, it was the Arpeggione Sonata that captured the incredible fusion of their musical sensibilities. For Olga, it says that life is beautiful, eternal and without boundaries. "You have two people, two instruments, but when you listen to it, it sounds just like one." I asked her if he had played it often. "No," she replied. "Once he had played it with Britten, he didn't want to play it with anyone else."

John Bridcut's film Rostropovich: The Genius of the Cello premieres at the Aldeburgh festival on 21 June, and will be shown on BBC4 in the autumn. This is an edited version of an article that will appear in the Aldeburgh festival programme.