Norris made her professional stage debut in a 1989 production of A Midsummer Night's Dream, which earned her her Equity card. She made her television debut in the 1991 BBC serial The Men's Room, playing the daughter of Bill Nighy's character. Other early television roles include appearances in Agatha Christie's Poirot, the television serial Clarissa, and a 1991 episode of Drop the Dead Donkey. She continued to make guest appearances in series such as Between the Lines and Casualty. When out of work, she supported herself by working on Sainsbury's supermarket checkouts, and selling double glazing in a shopping centre. In 1995 she played Helen in the Royal Exchange, Manchester's production of Look Back in Anger.

After being out of work for four months in 1996, Norris considered quitting acting and reading for a degree in law, intending to become a solicitor. However, she got a part in Cold Feet playing Karen Marsden, a middle-class woman who feels trapped by her middle-class lifestyle. Norris appeared in every episode and was nominated for a British Comedy Award for Best Actress in 2001.

In 2002, Norris began a relationship with Simon Wheeler, a Writer on Wire in the Blood. The couple married in December 2002 in a ceremony at the Tower of London. Their son, Wilf, was born in June 2004, and their daughter, Hero, was born in August 2007. Norris' father-in-law is General Sir Roger Wheeler, the Chief of the General Staff from 1997 to 2000.

At the end of 2005 she was cast in the BBC One spy drama Spooks, playing Ros Myers. She appeared throughout the 2006 series, then in eight of the ten episodes in the 2007 series before taking time off filming for maternity leave. She returned to the show for the 2008 series. For her part, she won the Best Actress award at the inaugural ITV3 Crime Thriller Awards. She was nominated in the same category the next year. She left the series in 2009 after four years.

From 2007 to 2009, she appeared in three series of Kingdom playing Beatrice Kingdom, the half-sister of Stephen Fry's character. She took the role as a change of pace from the "ice maiden" characters she often portrays. In 2010, she starred opposite Trevor Eve in the remake of Bouquet of Barbed Wire. In 2010, she was cast in the television science fiction drama Outcasts as Stella Isen, the head of security on an extraterrestrial human colony. Filming occurred on location in South Africa. From November 2010, Norris is due to play Ruth Condomine in a national tour of Noël Coward's Blithe Spirit. She will star opposite her Cold Feet co-star Robert Bathurst, and Alison Steadman. Of the play, Norris said, "Coward's insights into the way the female mind works are excellent. Elvira's manipulations and Ruth with her control issues, they're both utterly believable". She also starred in The Crimson Field a British Production about a field Hospital in France in WW1.

On 4 October 2014, Norris guest starred in Episode 7 of the 8th series of the BBC series Doctor Who, 'Kill The Moon'. She played Lundvik, an Astronaut.