Morrow was born in Chicago and raised in Des Moines, Iowa by parents who were both classical singers. As a teenager, she first heard recordings by actress/singer Susan Johnson, which inspired her to try musical theater, beginning with the role of Meg in Brigadoon.
After graduating from St. Joseph Academy (Des Moines, Iowa) in 1954, she graduated from Clarke College in 1958. She then moved to Milwaukee, Wisconsin, where she taught and performed on the side, but she soon moved to Manhattan. Her career there began with an appearance in the Off-Broadway show, Sing, Muse! in 1961; she won a 1962 Theatre World Award for her performance. She played Luce in a 1963 Off-Broadway revival of Rodgers & Hart's The Boys from Syracuse and a number of roles in City Center over the next five years, including The Most Happy Fella in 1966 as Cleo.
Morrow's first Broadway production, I Had a Ball, with Buddy Hackett and Richard Kiley, opened in December 1964, and folded six months later amid lackluster reviews. Most of Morrow's later Broadway productions would also have short runs despite some impressive casts. Two years later, she appeared as Mary Texas in A Joyful Noise, a misconceived effort to incorporate country music into a Broadway show. It closed after 4 previews and 12 performances. Next she played Na'Ama in 1968's I'm Solomon, which vanished just as quickly (9 previews, 7 performances). In November 1971, she appeared as Babylove in The Grass Harp (co-starring Barbara Cook, based on Truman Capote's novel of the same name), which lasted one week at the Martin Beck Theatre, but brought her good notices, especially for her rendition of "Babylove Miracle Show". In 1972, Morrow appeared as Irene Jantzen in another short-lived Broadway production, The Selling of the President, which had 5 regular performances. Her last Broadway performance was as The Princess Puffer/Miss Angela Prysock in The Mystery of Edwin Drood , as a replacement, in December 1986.
Asked why so many of the Broadway shows that she appeared in were flops, Morrow said, "I've analyzed this, trying to think of why I've had so many flops. I keep coming back to my contemporaries ... it was always the ones who could sing but also had something extra, something interesting about themselves ... I think with me, I was just a singer with a big voice and I was pleasant, and that can only take you so far."
From 1969, she made Los Angeles, California her home base, where she worked on local cable in a short-lived series called Singin' that co-starred her friend and peer, Nancy Dussault and appeared regularly on The Jim Nabors Hour (1969), The Merv Griffin Show, The Tonight Show and Match Game. She won an Emmy Award (with Nancy Dussault) for the PBS program "Cabaret Tonight". Other TV work has included appearances in many series: Girl Talk (1968), Love, American Style (1973/two episodes), Medical Center (1974), Paul Sand in Friends & Lovers (1974), Karen (1975/two episodes), Starsky & Hutch (1976), Tabitha (1977/Aunt Minerva), Alice (1979), Friends (1979/Pamela Richards), Ladies' Man (1980/Betty Brill), The Love Boat (1979 and 1982/two episodes), Too Close for Comfort (1983), Trapper John, M.D. (1983), Goodnight, Beantown (1984), Falcon Crest (1988/two episodes), Murder, She Wrote (1988-89/Gladys), Night Court (1989) and Sabrina, the Teenage Witch (1996-97/Nana).
Morrow has also appeared in three TV movies: Eve Wister in I Was a Mail Order Bride (1982), Martha Biggs in The Boy in the Plastic Bubble (1976), and Mrs. Turner in Cage Without a Key (1975).
She was also a regular on Garrison Keillor's A Prairie Home Companion.
Her stage work outside of New York has included Parthy in Show Boat (1997, national tour of the 1994 Broadway production), Die Fledermaus, It's a Bird...It's a Plane...It's Superman (1967 revival), Anything Goes, Annie Get Your Gun, Sally Adams in Call Me Madam (2000), Hello, Dolly!, Oliver!, Carlotta in Follies (1990 Long Beach Civic Light Opera) Sweeney Todd and A New Brain. She also has appeared in concerts with major orchestras in Los Angeles, Milwaukee, and Honolulu.
She can be heard on the 2002 concept album Miss Spectacular, a studio recording of an unproduced work by Jerry Herman, with whom she has worked and recorded extensively. Other albums include An Evening with Jerry Herman (1998).
Morrow taught musical theatre performance at UCLA and continues to teach master classes in performance and audition skills. She is a member of the faculty of AMDA.