At 23, she achieved her dream of becoming an Actress, graduating from the London Academy of Music and Dramatic Art (LAMDA). Through the 1980s, she became a dynamic stage Actress, both in London and on Broadway. She starred in the premiere of David Hare's The Secret Rapture, and won the first of her three Olivier Awards in 1995. In 1983, she starred with Ben Cross in the BBC's serial version of A.J. Cronin's The Citadel, playing the role of Christine Manson. For the big screen, Higgins played Julia Cotton in Clive Barker's Hellraiser (1987), based on Barker's novella The Hellbound Heart. She reprised the role for Tony Randel's Hellbound: Hellraiser II (1988). Her other film credits include the Scottish film Small Faces (1996), Woody Allen's Cassandra's Dream (2007), and The Golden Compass (2007).

A six-time Olivier Award nominee for her work in the theatre, Higgins received her first nomination in 1984 for her role as Stella Kowalski in A Streetcar Named Desire. She has since won the Olivier Award for Best Actress three times; for Sweet Bird of Youth in 1995, Vincent in Brixton in 2003 and Hecuba in 2005. She made her Broadway debut in 2003 in Vincent in Brixton, receiving a Tony Award nomination for Best Actress in a Play. She returned to Broadway in the 2014 revival of A Delicate Balance.

Higgins was awarded the Laurence Olivier Theatre Award in 1995 (1994 season) for Best Actress in a Play for her performance in Sweet Bird of Youth at the Royal National Theatre. She won the same award in both 2002 and 2005: in 2002 for her performance in Vincent in Brixton performed at the Royal National Theatre, Cottesloe and Wyndham's Theatres, and in 2005 for her performance as Hecuba in the Euripides tragedy at the Donmar Warehouse. She was awarded the 2002 London Critics' Circle Theatre Award for Best Actress for her performance in Vincent in Brixton. Additionally, she was awarded the 1994 London Critics Circle Theatre Award (Drama Theatre) for Best Actress for her performances in The Children's Hour and Sweet Bird of Youth. She was also awarded the 2002 London Evening Standard Theatre Award for Best Actress for her performance in Vincent in Brixton at the Donmar Warehouse in London. In 2003, she was nominated for a Tony Award as Best Actress in a Play for Vincent in Brixton on Broadway, and garnered the 2003 Theatre World Award for outstanding major Broadway debut.

From mid-May to September 2009, she appeared as the Countess Rossillion in All's Well That Ends Well at the National Theatre (Olivier stage).

On 30 September 2012, Higgins appeared in Season 3, episode 3 of Downton Abbey as Mrs Bartlett, a friend of Mrs Bates who eventually helps in the release of Mr Bates from prison.

On 14 November, 2013, she appeared in the Doctor Who mini-episode "The Night of the Doctor", which starred Paul McGann as the Eighth Doctor. Higgins played Ohila, the leader of the Sisterhood of Karn. Higgins returned to the role in 2015 in "The Magician's Apprentice", which originally aired on 19 September 2015, and in the series finale "Hell Bent" on 5 December 2015. The same year she appeared in EastEnders, as prosecuting Lawyer Hazel Warren.

In January 2017, Higgins appeared in CBBC's adaptation of Jill Murphy's The Worst Witch as Miss Ada Cackle, and her wicked twin sister Agatha.