Simpson was famous for his writing partnership with Ray Galton.
His manager Tessa Le Bars said: "Having had the privilege of working with Alan and Ray for over 50 years, the last 40 as agent, business manager and friend, and latterly as Alan's companion and carer, I am deeply saddened to lose Alan after a brave battle with lung disease."
Galton and Simpson met at Milford Sanatorium when they were both diagnosed with tuberculosis as teenagers.
Galton and his family paid tribute in a joint statement, saying: "There are no words to express our sense of loss and sadness at the passing of Alan Simpson, Ray's partner and family friend over the last 70 years.
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"From their first attempts at humour in Milford Sanatorium, through a lifetime of work together, the strength of Alan and Ray's personal and professional bond was always at the heart of their success. We respectfully request there are no attempts to contact the Galton family home at this time."
As well as their acclaimed work with Tony Hancock and Steptoe actors Harry H Corbett and Wilfrid Brambell, Galton and Simpson also wrote TV, film and stage scripts for the likes of Frankie Howerd, Peter Sellers, Leonard Rossiter, Arthur Lowe and Les Dawson.
After sending a skit to a BBC talent spotter, the pair got their first big break with Hancock's Half Hour, which began on radio in 1954 before moving to the small screen.
Next, in 1962, came the hugely popular sitcom Steptoe And Son, about father and son rag-and-bone men living together in a grimy house in Shepherd's Bush.
The sitcom became the most popular programme on TV and is seen as one of the shows from the Golden Age of the genre.
It earned Galton and Simpson, who had not been expected to survive tuberculosis in their teenage years, Writers' Guild Awards in 1962 and 1963.
In 1997, a six-part BBC series called Get Well Soon, written by Galton and John Antrobus, was based on Galton and Simpson's experiences in Milford Sanatorium.
Proving the continued popularity of their characters, the pair successfully revived Steptoe And Son for a play in 2005 called Murder At Oil Drum Lane.
Both were both awarded OBEs in 2000 for their contribution to British television.
The pair said last year that they still met every Monday to drink coffee and discuss the best comedians.
"When there were only two channels, there was a load of rubbish. A show might start out dreadful, but you nursed it," Simpson told the Daily Telegraph, rubbishing the suggestion that there had ever been a TV Golden Age.
"Fifty years ago, if you had an idea, it could be going out within three weeks - the time it took to build the sets. Now it has to go through committees, and the process can take years."
Galton and Simpson were honoured with the Bafta Fellowship last summer.
"We always wanted a Fellowship, even though we did not know what a Fellowship was. Not the sort of thing one associates with a couple of Cockney lads, apart from Alfred Hitchcock, of course," Simpson said at the time.