The 86-year-old triple Oscar and triple Bafta winner, who was born in Toronto, Canada and moved to Britain in the 1950s, died at his home in St Andrews, Bristol, on Friday, his family said.
His daughter Natasha Sutton Williams told the PA news agency that her father had been suffering from cancer, in what she said had been quite a swift illness.
Describing her "fabulous" father who had six children, she said: "He really was an inspiration to everyone that met him. Whether they were animators, or from the top to the bottom of society.
"An incredibly generous, warm-spirited person who really wanted to learn about the world."
Williams was the animation director on the 1988 blockbuster Who Framed Roger Rabbit? - creating characters including Roger and Jessica Rabbit.
The live-action animated film starring Bob Hoskins saw Williams win a Bafta as well as two Oscars - one in the special academy award category and one for special effects.
Williams also animated the title sequences for the 1970s comedy classics The Return Of The Pink Panther and The Pink Panther Strikes Again, and worked on Casino Royale.
Williams has previously credited Snow White - which he saw at the age of five - as having a "tremendous impression" on him.
On a visit to Disney at the age of 15, Ms Sutton Williams said her father waited at the gates and met all the lead animators who later taught him and became his friends.
Williams told the BBC in 2008: "I always wanted, when I was a kid, to get to Disney. I was a clever little fellow so I took my drawings and I eventually got in.
"They did a story on me, and I was in there for two days, which you can imagine what it was like for a kid."
After that he said he was advised to learn how to draw properly and admitted he "lost all interest in animation" until he was 23 - throwing himself into art.
He said he was drawn back to the craft because his "paintings were trying to move".
His first film, The Little Island, was released in 1958 and scooped a Bafta, and his animated adaptation of Charles Dickens' A Christmas Carol in 1971 saw him take home his first Oscar.
During his lengthy career, Williams also wrote a how-to book called The Animator's Survival Kit and was animating and writing until 6pm on the day he died, his daughter said.
She described her dad as the "link between the golden age of animation from the 1940s to the golden age of CGI and digital animation of now".
And she said her father had two studios - including one at his home where he had "several Disney animation desks", and had been using one used in Pinocchio.
His second studio was at Bristol-based Aardman Animations - the home of Wallace and Gromit.
"He had a fabulous work ethic... he had incredible vibrancy and flair, and was an actor at heart and essentially all of his animation was him acting through pictures," Ms Sutton Williams added.
"He was incredibly supportive to his family and to his friends, and to his children - he was my number one fan."
Williams is survived by his wife Imogen Sutton.