Copeland began her career in the theatre, making her professional debut as Juliet in william Shakespeare's Romeo and Juliet at the Brooklyn Academy of Music in 1945. She made her Broadway debut as Nadine in the original 1948 production of Bessie Breuer's Sundown Beach. Since then she has maintained an active career in the theatre. Her other Broadway credits include Detective Story (1949), Not for Children (1951), Handful of Fire (1958), Tovarich (1963), Something More! (1964), The Price (1968), Coco (1969), Two By Two (1970), Checking Out (1976), 45 Seconds from Broadway (2002), Wit & Wisdom (2003) among others. Copeland was nominated for a Drama Desk Award for the 1977 production of Pal Joey and won a Drama Desk Award in 1981 for The American Clock.

Copeland has also worked extensively Off-Broadway in New York City. Notable credits include Desdemona in Othello at the Equity Library Theatre (1946), Betty Shapiro in The Grass is Greener at the Downtown National Theatre (1955), Melanie in Conversation Piece at the Barbizon-Plaza Theatre (1957), Mrs. Erlynne in Delightful Season at the Gramercy Arts Theatre (1960), Leonie Frothingham in End of Summer at the Manhattan Theatre Club (1974), Lillian Hellman in Are You Now or Have You Ever Been at the Promenade Theatre (1978), the title role in Candida at the Roundabout Theatre (1979), Tasha Blumberg in Isn't It Romantic? at the Playwrights Horizons (1983), Mrs. Thompson in Hunting Cockroaches at the Manhattan Theatre Club (1987), Rose Brill in The Rose Quartet at the Circle Repertory Theatre (1991), Aida Gianelli in Over the River and Through the Woods at the John Houseman Theatre (1998), Nelly Fell in The Torchbearers at the Greenwich House Theatre (2000), and a part of a rotating cast in Wit & Wisdom at the Arclight Theatre (2003). She won an Obie Award in 1991 for her portrayal of Eva Adler in The American Plan at the Manhattan Theatre Club.

Copeland began working on television in the early 1950s as a guest Actress on such shows as Suspense and The Web and the live telecast of O'Neill's play The Iceman Cometh in 1960. She has appeared on numerous soap operas. She portrayed Andrea Whiting (Joanne's daughter, Patti's malevolent former mother in-law) on Search for Tomorrow, twin sisters Maggie and Kay Logan on Love of Life, and roles on The Edge of Night, How to Survive a Marriage and As the World Turns. She also portrayed Gwendolyn Lord Abbott on One Life to Live from 1978–1979, and later returned to the series to play Selma Hanen in 1995. Between 1993-1997 she portrayed the recurring character of Judge Rebecca Stein on Law & Order. Her other television credits include guest appearances on The Patty Duke Show, Chicago Hope, ER, All in the Family, and Naked City to name just a few.

Copeland made her first film appearance as Alice Marie in The Goddess (1958). Her film career has been sporadic and her appearances have been almost exclusively in prominent secondary roles. Her film credits include Middle of the Night (1959), Roseland (1977), It's My Turn (1980), A Little Sex (1982), Happy New Year (1987), The Laser Man (1988), Her Alibi (1989), Jungle 2 Jungle (1997), The Peacemaker (1997), The Object of My Affection (1998), The Adventures of Sebastian Cole (1998), The Audrey Hepburn Story (2000), The Last Request (2006), and The Private Lives of Pippa Lee (2009). She also voiced Tanana in Disney's Brother Bear (2003).

In December 2014, Copeland was invited, along with Broadway actor-singer Jamie Ross by The Noel Coward Society to lay flowers on the statue of Sir Noël Coward at The Gershwin Theatre in Manhattan to celebrate the 115th birthday of "The Master".