Law won the Ballon d'Or and scored 237 goals in 404 appearances for United.
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."
United also said in a statement: "He will always be celebrated as one of the club's greatest and most beloved players.
"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."