Alexander won the 1969 Tony Award for Best Featured Actress in a Play for her performance in the Broadway production of The Great White Hope.

[4] Alexander's major break in acting came in 1967 when she played Eleanor Backman in the original production of Howard Sackler's The Great White Hope at Arena Stage in Washington, DC.

Like her co-star, James Earl Jones, she went on to play the part both on Broadway (1968), winning a Tony Award for her performance, and in the film version (1970), which earned her an Oscar nomination.

The play The Time of Your Life was revived on March 17, 1972, at the Huntington Hartford Theater in Los Angeles with Alexander, Henry Fonda, Gloria Grahame, Lewis J. Stadlen, Richard Dreyfuss, Ron Thompson, Strother Martin, Richard X. Slattery, and Pepper Martin among the cast with Edwin Sherin directing.

[8] Alexander's other television films include Arthur Miller's Playing for Time, co-starring Vanessa Redgrave, for which Alexander won another Emmy Award; Malice in Wonderland (as famed gossip-monger Hedda Hopper); Blood & Orchids; and In Love and War (1987) co-starring James Woods, which tells the story of James and Sybil Stockdale during Stockdale's eight years as a US prisoner of war in Vietnam.

The show's frank portrayal of "senior" sexuality and explicit sex scenes generated controversy, although it won a rare endorsement by the AARP.

In 1993, President Bill Clinton appointed Alexander chairperson of the National Endowment for the Arts, the organization that had provided partial funding for The Great White Hope at Arena Stage.

Alexander had been acting regularly in various regional theaters when she met producer/director Edwin Sherin in Washington, DC, where he was artistic director at Arena Stage.

Alexander starred in the original theatrical production of The Great White Hope under Sherin's direction at Arena Stage prior to the play's Broadway debut.