His death was announced to the post-punk groups's fans on Twitter on Saturday.
The guitarist was described as "one of the best ever to do it", and "a bit of a genius" by his bandmates.
The statement added: "This is so hard for us to write, but our great friend and supreme leader has died today.
"Andy's final tour in November was the only way he was ever really going to bow out; with a Stratocaster around his neck, screaming with feedback and deafening the front row.
"His uncompromising artistic vision and commitment to the cause meant that he was still listening to mixes for the upcoming record, whilst planning the next tour from his hospital bed."
The band added that they would "remember him for his kindness and generosity, his fearsome intelligence, bad jokes, mad stories and endless cups of Darjeeling tea".
In 2019, Gang Of Four embarked on a 40th anniversary world tour, playing in countries including Japan, Greece, Spain, Australia and New Zealand.
Gill's wife Catherine Mayer tweeted: "This pain is the price of extraordinary joy, almost three decades with the best man in the world."
As well as performing, Gill was also a producer and worked with bands including the Red Hot Chili Peppers and The Stranglers.
The post-punk band formed in the city in 1976 and their debut album, Entertainment!, was voted the fifth greatest punk album of all time by Rolling Stone magazine.