The iconic Scotland striker, nicknamed 'Yogi', played up front for the Hoops during a glittering era in the 1960s.
Hughes was awarded a European Cup winners' medal after helping Celtic reach the 1967 final, although injury ruled him out of the victory over Inter Milan in Lisbon.
He remains the club's eighth-highest goalscorer in history afrer a 12-year spell and also played for Sunderland and Crystal Palace during the latter stages of his career.
Celtic said in a statement: "He was dearly loved and will sorely be missed. Always remembered and celebrated with love by his wife Theresa, his sons Kevin, Martin and John, his daughter Joanna, his six grandchildren and extended family.
"He now joins the ranks of those who will be immortal as long as there's a Glasgow Celtic."
Hughes' most famous contribution came in the 1970 European Cup semi-final against Leeds.
After a 1-0 victory for Celtic in the first leg, the Whites drew level in the return fixture at Hampden Park, before a header from Hughes swung the momentum in favour of the Hoops and led to a 2-1 victory.
Around 140,000 people attended the match, a record for a European fixture which is unlikely to be broken.
Among them was boyhood Celtic fan Alan Brazil, who recalls a night he will never forget.
He told talkSPORT: "Let me take you to that game at Hampden, the semi-final.
"We skipped in. We got in really early. We climbed over and hid until the gates opened.
"It was unbelievable, the momentum bulding, the noise, and Celtic won 2-1.
"Yogi got the equaliser, what a game that was! I remember how I felt walking home, I will never forget that, it was magic."