Carnelia started his career in theatre as an actor. He played The Boy (Matt) in the Off-Broadway production of The Fantasticks. He left the production to pursue songwriting and musical theatre composition. His songs were first heard, along with the work of other writers, in Working. He then wrote the full score for the Broadway musical is there life after high school. After high school, he composed the score of Three Postcards, which, in addition to being produced at Playwrights Horizons, was revived at the Circle Repetory Theater. He played Bill in the PH production of Three Postcards. Carnelia went on to collaborate with Marvin Hamlisch on Sweet Smell of Success. He also wrote songs for Nora Ephron's play Imaginary Friends.
Throughout his career, Carnelia has been nominated for and has won several awards. He has been nominated for three Tony Awards and two Drama Desk Awards. He has won the Johnny Mercer Award, the Gilman and Gonzalez-Falla Musical Theater Award, and the Kleban Award.