He received a degree in composition from the Eastman School of Music and studied with Aaron Copeland and Nadia Boulanger. He started his professional music career as a piano player for dance bands. Later he became a pianist for theatre rehearsals.
Bye Bye Birdie, his first and probably most-known success, was responsible for winning him his first Tony Award. The London production of the show won the London Critics' Best Foreign Musical Award. Strouse also co-wrote the 1966 musical It's a Bird, a Plane, It's Superman and Golden Boy starring Sammy Davis Jr. In 1970, Strouse won another Tony for Applause starring Lauren Bacall.
Two of the musicals were made into major motion pictures: Bye Bye Birdie, starring Janet Leigh, Dick Van Dyke, and Ann Margaret (Columbia, 1963), and Annie (Columbia, 1982). The soundtrack to Annie went gold -- extremely rare for a show tunes album -- mainly off of the ballad "Tomorrow." The song has been covered by numerous artists including Johnny Mathis and Cissy Houston. The LP has the distinction of winning both a Tony and two Grammy awards. Bye Bye Birdie was also a 1995 ABC-TV movie starring Jason Alexander of Seinfeld. The song "Let's Settle Down" won a 1996 Primetime Emmy Award.
Strouse contributed scores and or songs to the following movies: Bonnie and Clyde (1967), The Night They Raided Minsky's (1968), There Was a Crooked Man (1970), Just Tell Me What You Want (1980), Ishtar (1987), and All Dogs Go to Heaven (1989).
Other musicals Strouse has contributed to include Broadway Musical, Charlie and Algernon, All American written with Mel Brooks and Lee Adams, Mayor, Dance a Little Closer, Nick and Nora, Annie Warbucks, Rags, and Nightingale. By Strouse, a musical consisting of songs from Strouse's catalog, tours frequently, while Strouse himself is an active guest lecturer.