Sallis is best known for his role as the main character Norman Clegg in the long-running British TV comedy Last of the Summer Wine, set in a Yorkshire town. He is the longest serving cast member, and the only one surviving from the programme's first episode in 1973.

He is also famous for providing the voice of Wallace in the Wallace and Gromit films, utilising another northern accent.

Sallis was born on 1 February 1921 in Twickenham (then Middlesex now Greater London), England. After attending Minchenden Grammar School in North London, Sallis went to work in a bank. After the outbreak of World War II he joined the RAF. He failed to get into aircrew because he had a serum albumin disorder and he was told he might black out at high altitudes. He became a wireless mechanic instead and went on to teach radio procedures at RAF Cranwell.

Sallis started as an amateur actor during his four years with the RAF when one of his students offered him the lead in an amateur production. His success in the role caused him to resolve to become an actor after the war, and so he trained at the Royal Academy of Dramatic Arts, making his first professional appearance on the London stage in 1946.

Sallis became a notable character actor on the London stage in the 1950s and 1960s, appearing alongside theatrical legends such as Laurence Olivier, John Gielgud, Ralph Richardson, Orson Welles, Judi Dench and Patrick McGoohan. He also appeared in character parts in British films of the time, including a few Hammer Films. In 1968, he was cast as the well intentioned Coker in a BBC Radio production of John Wyndham's The Day of the Triffids.

His first notable television role was as Samuel Pepys in the BBC serial of the same name in 1958. In 1961, he appeared as Gordon in the "Find and Destroy" episode of Danger Man. He appeared in the Doctor Who story "The Ice Warriors" in 1967, playing renegade scientist Elric Penley; and in 1983 was due to play the role of Striker in another Doctor Who story, "Enlightenment" before having to withdraw.

In 1970, he was cast in the BBC comedy series The Culture Vultures, which saw him play stuffy Professor George Hobbs to Leslie Phillips' laid-back rogue Dr Michael Cunningham. During the production, Phillips was rushed to hospital with an internal haemorrhage and as a result, only five episodes were ever made.

Sallis was cast in a one-off pilot for Comedy Playhouse entitled "Funerals and Fish" (1973), which became Last of the Summer Wine, as the unobtrusive lover of a quiet life, Norman Clegg. Sallis had already worked with Michael Bates, who played unofficial ring-leader Blamire in the first two series, on stage. The pilot was successful and the BBC commissioned the series.

Sallis is still playing the role of Clegg, and is one of only two cast members remaining from the original Comedy Playhouse pilot, along with Jane Freeman who plays Ivy, the cafe owner. In 1988 he appeared as Clegg's father in First of the Summer Wine, a prequel to Last Of The Summer Wine set in 1939. In 1974 he played Mr Bonteen in the BBC period drama The Pallisers. Between 1976 and 1978 he appeared in the children's series The Ghosts of Motley Hall, in which he played Mr Gudgin, an estate agent who did not want to see the eponymous Hall fall into the wrong hands. In 1977 he played Rodney Gloss in the BBC series Murder Most English.

In 1978, he starred alongside Northern comic actor David Roper in the ITV sitcom Leave it to Charlie as Charlie's pessimistic boss. The programme lasted for four series, ending in 1980. Also in 1978, he played the part of the ghost hunter Milton Guest in the children's paranormal drama series The Clifton House Mystery.