Born Siobhán Giollamhuire Nic Cionnaith in Belfast, Northern Ireland, she grew up in Galway City and in County Monaghan, Ireland speaking fluent Irish. She was still in her teens when she became a member of an amateur Gaelic theatre group and made her stage debut at Galway's Gaelic Theatre in 1940. She is also remembered for her English-language performances at the Abbey Theatre in Dublin where she would eventually star in what many consider her finest role in the George Bernard Shaw play, Saint Joan. While performing at the Abbey Theatre she met actor Denis O'Dea, whom she married in 1956 and they had one child, a son: Donnacha O'Dea, who swam for Ireland at the 1968 Summer Olympics and later won a World Series of Poker bracelet in 1998.

In 1947 she debuted on the London stage and on Broadway in 1955 in The Chalk Garden for which she would receive a Tony Award nomination for "Best Actress in a Leading Role, Drama." In 1956, she appeared in the Cambridge Drama Festival production of Saint Joan at the Off-Broadway Phoenix Theatre. Theatre critic Elliot Norton called her performance the finest portrayal of Joan in memory. Siobhán McKenna's popularity earned her the cover of Life magazine. She received a second Tony Best Actress nomination for her role in the 1958 play, The Rope Dancers in which she starred with Art Carney and Joan Blondell.

Although primarily a stage actress, McKenna appeared in a number of made-for-television films and dramas plus acted in several motion pictures including 1961's King of Kings, starring in the role of the Virgin Mary. In 1964 she performed in Of Human Bondage and the following year in Doctor Zhivago.

McKenna was awarded the Gold Medal of the Eire Society of Boston, Massachusetts for having "significantly fulfilled the ideals of the Eire Society, in particular, spreading awareness of the cultural achievements of the Irish people."

Siobhán McKenna's final stage appearance came in the 1985 play Bailegangaire for the Druid Theatre Company. Suffering from lung cancer, despite surgery, she died the following year in Dublin, Ireland at the age of 64, and was interred in the Rahoon Cemetery in County Galway, Ireland. The inscription on her tomb is written in Irish.