During the 1960s, he won the Sibelius and Bach Competitions, while also placing in the top-five of the Enescu and Tchaikovsky Competitions. Upon Kuznetsov's death, Kagan began studying with David Oistrakh, and in 1969, he began playing chamber music with Richter. Along with Richter and Gutman, Kagan also appeared frequently with pianistVasily Lobanov, who would later dedicate a piece to him.
As Kagan seemed to be approaching the zenith of his career he became seriously ill with cancer in 1989. He had several surgeries, but struggled to remain active, touring Europe when he could and arranging festivals. Though his doctors at a hospital in Lübeck, Germany declared him too sick to be released, Kagan discharged himself to appear at his final festival, in Kreuth am Tegernsee, Bavaria. Shortly after giving two Mozart concerts there, where he had to be helped on-stage, he died on 15 July 1990, aged 43.