Sutton has appeared in many films during his career, including Rotten to the Core (1965), Crossplot (1969), The Devils (1971), Madame Sin (1972), The Pink Panther Strikes Again (1976), Fellini's Casanova (1976), Edward II (1991), and The Football Factory (2004).
Sutton became known after playing a gay biker in The Leather Boys (1964), a role which showed his potential for eccentric screen personae. He married American Actress Marjorie Steele in 1961; she had previously been married to the millionaire Producer Huntington Hartford. Sutton and Steele had one child together, but divorced in 1965. On stage, he played the title role in the first production of Joe Orton's Entertaining Mr Sloane (1964). From 25 May 1966 he appeared in Tango, a play by Slawomir Mrozek at the Aldwych Theatre alongside Patience Collier, Peter Jeffrey, Mike Pratt, and Ursula Mohan under Director Trevor Nunn.
Among his many television appearances are his roles as Tinker Dill in Lovejoy (1986 & 1991 – 94) – whose friendship with Lovejoy, the title character, and expertise in the antique trade was the backbone of the show – as Mr Carter in the Beiderbecke Trilogy and as Oleg Kirov in Smiley's People (1982). He also featured in The Sweeney episode 'Golden Boy' and in a Christmas special episode of Porridge (1976) as the somewhat unstable trusty-turned-hostage-taker Reg Urwin, with Ronnie Barker and Richard Beckinsale.
In 1999 he appeared in the BBC Radio play Cosmos the Mystic Dog. In 2004 he made an appearance in the soap opera EastEnders for sixteen episodes, playing Wilfred Atkins, a conman.
In 2003, Sutton found inspiration from the internet "where apparently people say that every time you masturbate God kills a kitten". From that statement, he developed a comic piece about "a young man's emotions and feelings, from the moment he's a baby tugging at his cock onwards." In August 2003, he performed the one-man Killing Kittens show at Edinburgh's Underbelly. Sutton followed up Killing Kittens with a second autobiographical show Pandora's Lunchbox in 2006. Following an acclaimed performance as william Blake in Peter Ackroyd's BBC television series The Romantics, Sutton joined the cast of Albion Rising at St. Giles in the Fields Church, London, in April 2007.
Sutton had a small role in the British teenage drama Skins as Freddie's granddad. He also appeared in the episode of Holby City broadcast on 15 March 2011 as a patient who fell down an escalator in a shopping centre. In 2012 he featured in the video "Once And For All" by Clock Opera.
Sutton also appeared in episode three of the BBC comedy series Family Tree ("The Austerity Games"), which aired in July 2013, and guest-starred in episode three of the BBC series Boomers in 2014. He played william Makepeace in Emmerdale in 2014.
In 2015 he appeared as a Roman Catholic Rector in the BBC TV series Father Brown episode 3.6 "The Upcott Fraternity". He also appeared in three episodes of the BBC's day time show, Doctors, in August 2015.
In November 2017 he played the lead role in a video for the Tom Chaplin song Midnight Mass.
