Nagrin studied with Martha Graham, Anna Sokolow, Hanya Holm, and Helen Tamiris whom he later married. In addition to work as a modern dancer, Nagrin also performed on Broadway in Plain and Fancy, Up in Central Park, and Annie Get Your Gun, among other musicals.
In June 1954 he formed the Dance-Percussion Trio with David Shapiro and Ronald Gould (who would go on to form the New York Percussion Trio), and the group toured the United States in June and July of that year. Nagrin and his wife formed the Tamiris-Nagrin Company in 1960. When Tamiris died in 1966, Nagrin concentrated on a solo career. In the early 1970s Nagrin formed "The Workgroup", a performance company with a focus on improvisation.
Nagrin's better known choreographic works include: "The Peloponnesian War", "Strange Hero", "Man of Action", "Spanish Dance", and "Jazz, Three Ways". Additionally, he choreographed for the 1954 film His Majesty O'Keefe.
In 1985, a 15-hour compilation of Nagrin's work, The Nagrin Videotape Library of Dances, was assembled. It is held in the Dance Collection at the New York Public Library for the Performing Arts.
Nagrin, who holds a B.S. in Education from City College in New York, has taught at C.W. Post College, New York University, the American Dance Festival, and Arizona State University. He is the author of several books including "How to Dance Forever: Surviving Against the Odds" (1988) and "Choreography and the Specific Image" (2001).