Sosanya has appeared in many roles in the theatre, on television and in films. Her first big break in theatre was in Anthony and Cleopatra at the National Theatre, and with the 2001 series Teachers. She also appeared in Sorted, People Like Us, Love Actually, Nathan Barley, Renaissance, Casanova, as Karen Blaine in the Jonathan Creek episode "The Three Gamblers", Much Ado About Nothing, Cape Wrath/Meadowlands, the Doctor Who episode "Fear Her", and FM.

In 2003, she played Rosalind in As You Like It with the Royal Shakespeare Company ("RSC") at the Swan Theatre, Stratford-upon-Avon; in 2008 she returned to the RSC to play Rosaline in Love's Labour's Lost, and in 2009 appeared in a radio adaptation of a story from the short story collection The State of the Art.

In January 2010, Sosanya appeared as Mae Pollock in Tennessee Williams' play Cat on a Hot Tin Roof, at the Novello Theatre, London.

Sosanya played the character Alibe Silver in Treasure Island (2012). She has played a main character, Kate McKenzie, in series one to three of the BBC original drama Last Tango in Halifax during 2012–2015, and a main character, Lucy Freeman, in the TV series W1A in 2014. In 2015, she starred with Catherine Tate and Mark Gatiss in a new play, The Vote, in the run-up to the UK 2015 General Election. In 2016, she started in the play Young Chekhov trilogy and as DCI Laura Porter in the ITV series Marcella.

In 2014 she appeared in the world premiere production of Privacy at the Donmar Warehouse, London. In 2016 she appeared in the world premiere production of Elegy at the Donmar Warehouse.