Robinson attended a child agency school and made her television debut in The Cops at the age of 17. Following this she spent three years at the Webber Douglas Academy of Dramatic Art, where she was awarded a Laurence Olivier bursary from the Society of London Theatre. She had a small role as the Jamaican girl in the 2004 film Vera Drake but was given an entire character backstory by the Director Mike Leigh.
Her first stage role was in the highly lauded National/Complicite production of Measure for Measure. Following this she was part of the Gunpowder season at the Royal Shakespeare Company where she played the tortured servant May in Rupert Goold's production of Speaking Like Magpies by Frank McGuinness. In 2006 she played Eve in an acclaimed production of Paradise Lost. Commenting on her nude scene, Sheila Tracy of The Stage wrote "one can not imagine the action working in any other way". Her performance in Sugar Mummies that same year led New York to list her as one of "London's hottest young stage actors" in 2007.
In 2011 Robinson appeared in Philip Ridley's Tender Napalm for which she received Best Female at The Offie Awards. She then went on to play Ophelia in The Young Vic's production of Hamlet for which she received the Clarence Derwent Award. Her role as Ophelia led to her being included in The New York Times Magazine article 'Titans Of The Stage'.
In 2009, she played Josie Porritt in the BBC television series Hope Springs and appeared as newly qualified English Teacher Helen Hopewell in eight episodes of Waterloo Road's fifth series. From December 2009 to January 2010, she starred as the maid Florence in the Hampstead Theatre's Darker Shores. In a Daily Telegraph review that rated the play only two stars, Charles Spencer commended Robinson's performance as having "a warmth and emotional openness". Kate Bassett for the Independent on Sunday also praised her, writing that she "copes admirably". In 2010, she starred as police sergeant Sally Donovan in the BBC series Sherlock, continuing her role in the second and third series that aired in January 2012 and December 2013/January 2014.
In 2016, she appeared in "Hated in the Nation", an episode of the anthology series Black Mirror.
