The comedian and musician died of natural causes Sunday night, agent Nigel Morton said.
Innes was known for his work with the British comedy group Monty Python and The Bonzo Dog Band.
A statement released on behalf of his family Monday said: "It is with deep sorrow and great sadness that we have to announce the death of Neil James Innes on December 29, 2019.
"We have lost a beautiful, kind, gentle soul whose music and songs touched the heart of everyone and whose intellect and search for truth inspired us all," the statement said. "He died of natural causes quickly without warning and, I think, without pain."
Innes wrote songs for the popular film Monty Python And The Holy Grail, appeared in Monty Python's Life of Brian and toured the UK and Canada with the group.
He was also a member of the band The Bonzo Dog Doo Dah Band, later renamed the Bonzo Dog Band, whose song I'm The Urban Spaceman won him an Ivor Novello award.
Colleagues have been paying tributes to Innes on Twitter, including the prominent British actor Mark Gatiss, who fondly wrote "Sweet dreams, sweet idiot."