Shaw was a key player in the historic success of Villa, contributing to their triumph in the 1981 league title and their memorable victory in the European Cup in 1982 as part of a legendary team.
"Aston Villa Football Club is deeply shocked and profoundly saddened to learn that Gary Shaw, one of our European Cup-winning heroes, has passed away," the club said on Monday.
"Gary was one of our own, a talented striker who delighted supporters with his goalscoring exploits which helped fire Villa to success in the 1980s. Individual accolades would also follow for a player who was idolized by many on the terraces.
"He passed away peacefully earlier today surrounded by his family, who asked Aston Villa to release a statement on their behalf.
"The thoughts of everyone at the club are with Gary's family and loved ones at this extremely difficult time."