Morris was born in New Jersey. He has had a long career of composing music for Mel Brooks, starting in 1968 with The Producers which was Brooks’ first film. Morris continued to write the scores and songs for most of Brooks’ films. However, the music of Robin Hood: Men in Tights and Dracula: Dead and Loving It were both composed by Hummie Mann. In an interview with Film Score Monthly, Brooks explained that Morris couldn't do the music for Men in Tights or Dead and Loving It due to other commitments, but they're still friends. Morris also composed the music to To Be or Not to Be, a film in which Brooks starred as well as wrote but did not direct. The original music for The Elephant Man, a film that was produced by Brooks, was also scored by Morris.
According to Brooks on an audio commentary of Young Frankenstein, Morris’ best scores written for his movies were The Elephant Man and Young Frankenstein.