Braben's father was a butcher at St John's Market, Liverpool, and he was born in Monkswell Street, Dingle. He was evacuated to Anglesey as a child during World War II, and was a fan of radio comedy, particularly Arthur Askey. He left school in 1945 and worked in a tobacco factory before national service in the Royal Air Force, during which he was posted to the kitchen at RAF Kenley. He then worked as a market trader with his own greengrocery stall, writing jokes in his spare time.
Although shy, he sent jokes to various comedians that were appearing in Liverpool. His first was sold to Charlie Chester for 2s 6d (12½p), but his first major success was with Ken Dodd, with whom he worked for 12 years.
Braben's biggest success came when the BBC lured Morecambe and Wise from ITV. Bill Cotton, then in charge of Light Entertainment at the BBC, was looking for a writer following the duo's split from Dick Hills and Sid Green. The first Braben-penned Morecambe & Wise Show was broadcast in July 1969, and he wrote most of their BBC shows after that, including many of the Christmas specials. In 1980 he joined Thames Television to continue writing for the duo following their move back to ITV two years previously.
Braben, along with Morecambe and Wise, won the Society of Film Television Artists 1973 award for Outstanding Contribution to Television. He also won the Best British Light Entertainment Script award from the Writer's Guild of Great Britain in 1969, 1970, 1971 and 1973.
Braben wrote and appeared in various radio comedy shows for the BBC, including The Show With Ten Legs and The Worst Show on the Wireless. In 2001, Braben collaborated with Hamish McColl and Sean Foley to write The Play What I Wrote, a stage play and tribute to Morecambe and Wise, which opened at the Liverpool Playhouse Theatre before transferring to London's West End. The director was Kenneth Branagh.
Braben was married to Deidree an ex George Mitchell singer/dancer, had three children and six grandchildren.
Braben's autobiography, The Book What I Wrote, was published in 2004.