The four-part programme, Allen v Farrow, also contains new material such as interviews with people who knew the family.
Allen and wife Soon-Yi Previn issued a statement as the first episode was broadcast in the US, saying the documentary’s makers “had no interest in the truth”.
“Instead, they spent years surreptitiously collaborating with the Farrows and their enablers to put together a hatchet job riddled with falsehoods,” said the statement, issued via a representative.
“Woody and Soon-Yi were approached less than two months ago and given only a matter of days ‘to respond’. Of course, they declined to do so.”
Allen did not participate in the programme but audio excerpts were used from his memoir, Apropos Of Nothing.
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The statement added: “As has been known for decades, these allegations are categorically false.
“Multiple agencies investigated them at the time and found that, whatever Dylan Farrow may have been led to believe, absolutely no abuse had ever taken place.
“It is sadly unsurprising that the network to air this is HBO – which has a standing production deal and business relationship with Ronan Farrow. While this shoddy hit piece may gain attention, it does not change the facts.”
Relatives, investigators, witnesses and family friend Carly Simon feature in the documentary.
The director has long said Mia Farrow made up the claim and planted them in her daughter’s mind after learning that Allen was having an affair with the actress’s then-22-year-old adopted daughter Previn – now his wife.
Allen has never been arrested or prosecuted over the allegation, first made in 1993.
Thank you to everyone for their kind words, the outpouring of support means more to me than I can say ♥️ Speaking the truth is so difficult, but I hope any fellow survivors who watched last night know they are not alone. The truth is something that cannot be changed.