Elman was born in Philadelphia, Pennsylvania, United States,[1] but his family settled in Atlantic City, New Jersey when he was four.

In 1936, Elman joined the Benny Goodman orchestra as a trumpeter,[1] after playing briefly with a band led by Alex Bartha[2] at Steel Pier in Atlantic City, where Goodman heard him.

"[3][4] After leaving Goodman in 1940, Elman joined Tommy Dorsey[2] and stayed until he was drafted in 1943.

In 1956, he was asked to recreate a klezmer solo with the vocalist Martha Tilton for the movie, The Benny Goodman Story, but was unable to, his technique having since withered away.

Elman appeared performing it in the film, but another trumpeter, Mannie Klein, played the solo on the soundtrack.

He died in 1968 at the age of 54 and was buried at Mount Sinai Memorial Park Cemetery in Los Angeles.