Born in Chicago, Illinois, Kostal opted not to attend college, instead teaching himself musical arranging by studying the symphonic scores available at his local library. His first professional job was as a staff arranger for Design for Listening, an NBC radio show based in his hometown. Irwin was one of four children born to James and Emma Kostal. His siblings James, Jerome and Violet all remained in the Chicago area.[1]

After moving to New York City, Kostal was hired for Sid Caesar's popular variety series Your Show of Shows, and followed this with a stint at The Garry Moore Show. In the latter part of the decade he began working on Broadway, orchestrating for Shinbone Alley, The Music Man, Fiorello!, and West Side Story. His work on the latter resulted in his being hired to score the 1961 screen adaptation with Saul Chaplin, Johnny Green, and Sid Ramin. The quartet won both an Oscar and a Grammy Award for their contributions. Kostal later went on to collaborate with Saul Chaplin for the 1965 film adaptation of the Rodgers & Hammerstein musical The Sound of Music and won an Oscar for Best Music Scoring.