The Scottish icon - the only Scot to ever win the Ballon d'Or - scored 237 goals in 404 appearances during his time in the red side of Manchester, placing him third in United's history behind Wayne Rooney and Bobby Charlton.
In 2021, he was diagnosed with Alzheimer's disease and vascular dementia.
A family statement said: "It is with a heavy heart that we tell you our father Denis Law has sadly passed away. He fought a tough battle but finally he is now at peace.
"We would like to thank everyone who contributed to his wellbeing and care, past and much more recently.
"We know how much people supported and loved him and that love was always appreciated and made the difference. Thank you."
His former club, Manchester United - where he is part of a three-man statue outside Old Trafford with fellow icons Charlton and George Best - said Law will always be remembered as one of the club's all-time greats.
"He will always be celebrated as one of the club's greatest and most beloved players," the statement from United said.
"The ultimate goalscorer, his flair, spirit and love for the game made him the hero of a generation.
"Our deepest condolences go out to Denis' family and many friends. His memory will live on forever more."
Beginning his career at Huddersfield, he was signed by Manchester City four years later for an estimated transfer fee of £55,000, at the time a new British record.
Law moved to Torino in Italy a year later but failed to settle, eventually moving to Manchester United in 1962 for £115,000, setting another British record transfer fee in the process.
Success soon followed during one of the great periods in the Red Devils' history.
Nicknamed 'The King' and 'The Lawman' by fans and 'Denis The Menace' by opponents, Law helped his club win the old First Division in 1965 and 1967, the FA Cup in 1963, and two charity shields.
He was unfortunate to miss United's 4-1 European Cup final triumph over Benfica in 1968 through injury - the first time an English side won the competition. His 46 competitive goals in a single season remain a Manchester United record.
On the international stage, Law played for Scotland a total of 55 times and still jointly holds the Scottish international record goal tally with 30 goals alongside Kenny Dalglish.
Law was integral to his nation and was key in helping Scotland win the old British Home Championship six times.
He is the only person from Scotland to win the Ballon d'Or, which he achieved in 1964.
Before calling time on his career, Law made a controversial switch back to Manchester City for a final season, scoring nine goals in 24 matches, making his total tally for City 30 goals in 68 appearances across two spells.
After retiring from football, Law became a television pundit, patron of UK-based charity Football Aid, and established the Denis Law Legacy Trust, and he was appointed a CBE in the 2016 New Year Honours for services to football and charity.
'Fantastic man'
Former United and England captain Bryan Robson, who worked with Law in an ambassadorial role at Old Trafford, described the Scot as a "fantastic man".
"We have lost one of football's giants both as a player and a gentleman. It's incredibly sad," Robson wrote in a column in the Daily Mail.
"Denis was more than just a fantastic footballer, he was a fantastic man. So generous with his time and everything delivered with that great sense of humour of his."
Former Scotland international Joe Jordan said Law was a "proper icon".
"Denis was a special player and to lose someone like him, I think will affect an awful lot of Manchester United fans," Jordan told the BBC.
"A proper icon for his ability on the field to score goals but he had a special edge to him, not just as a player but as a person. He was a special player."
In an 18-year career, Law also played for Huddersfield, Manchester City (twice) and Torino.
City, the current English champions, said the "whole of Manchester" was mourning.
In a reply to United's X post breaking the news, they said: "The whole of Manchester, including everyone at City, is mourning with you."