Actor and comedian Jones, who directed some of the comedy troupe's best-loved works, "gently slipped away" at the age of 77, a statement to the PA news agency on behalf of his family said.
Jones's fellow former Python member Eric Idle tweeted that Jones's death "is a cruel and sad thing", adding: "But let's remember just what joy he brought to all of us."
Idle said he loved Jones "the moment I saw him on stage at the Edinburgh Festival in 1963.
Terry Gilliam said Jones, who had dementia and died on Tuesday evening, could not have been a better friend.
Writing on Twitter, he said: "HE WAS A VERY NAUGHTY BOY!!...and we miss you.
"Terry was someone totally consumed with life.. a brilliant, constantly questioning, iconoclastic, righteously argumentative and angry but outrageously funny and generous and kind human being... and very often a complete pain in the ass.
"One could never hope for a better friend.
Jones was remembered by other Python stars including Sir Michael Palin, who said Jones was "one of my closest, most valued friends".
Sir Michael said Jones was "kind, generous, supportive and passionate about living life to the full".
"He was far more than one of the funniest writer-performers of his generation, he was the complete Renaissance comedian - writer, director, presenter, historian, brilliant children's author, and the warmest, most wonderful company you could wish to have," he said.
He added in a statement to PA: "I feel very fortunate to have shared so much of my life with him."
John Cleese tweeted: "It feels strange that a man of so many talents and such endless enthusiasm, should have faded so gently away..."
Referring also to sixth Python member Graham Chapman, who died in 1989, he added: "Two down, four to go."
Jones appeared in the Monty Python's Flying Circus TV series in a variety of characters, and was known for his performances in drag.
A statement on behalf of his family said they were "deeply saddened" to announce the death of a "beloved husband and father", who had "given pleasure to countless millions across six decades".
They hailed an "extraordinarily talented, playful and happy man living a truly authentic life, in his words 'Lovingly frosted with glucose'."
Jones died "with his wife Anna Soderstrom by his side after a long, extremely brave but always good- humoured battle with a rare form of dementia, FTD", the statement said.
"Over the past few days, his wife, children, extended family and many close friends have been constantly with Terry as he gently slipped away at his home in north London.
"We have all lost a kind, funny, warm, creative and truly loving man whose uncompromising individuality, relentless intellect and extraordinary humour has given pleasure to countless millions across six decades."
They said his work "will live on forever, a fitting legacy to a true polymath".
The family thanked "wonderful medical professionals and carers for making the past few years not only bearable but often joyful. We hope that this disease will one day be eradicated entirely."
The Welsh-born star suffered from a form of dementia which affects the ability to communicate.
Jones directed the film Monty Python And The Holy Grail in 1975 with Terry Gilliam.
The group's Life Of Brian film in 1979, about a hapless man mistaken for Jesus, was attacked as blasphemous at the time but has since been voted the funniest classic comedy.
On screen, Jones's much-loved characters included Arthur "Two Sheds" Jackson, Cardinal Biggles of the Spanish Inquisition, and Mr Creosote, the monstrously obese restaurant patron.
Stephen Fry, David Walliams and Charlie Brooker were also among those paying tribute to the star, with Fry praising Jones as a "wonderful talent, heart and mind".
Walliams wrote: "Thank you Terry for a lifetime of laughter."
Shane Allen, BBC controller of comedy commissioning, wrote that it was a "sad day to lose an absolute Titan of British comedy" and "one of the founding fathers of the most influential and pioneering comedy ensembles of all time".
Jones also directed The Meaning Of Life in 1983, the Pythons' last film together.
His other credits include The Wind In The Willows in 1996, with performances from Idle, Palin and Cleese, 2015 comedy Absolutely Anything, and Personal Services (1987).
Jones took part in a reunion of remaining Monty Python members in 2014.
He had two children with Alison Telfer, whom he married in 1970, and became a father again, at the age of 67, with second wife Ms Soderstrom.