The St Helens-born winger died peacefully at his home at the age of 74.
The former England international joined the Reds in April, 1964 and lifted the league championship in his first full season at Old Trafford. After joining Burnley as a teenager, he became a real favourite at Turf Moor and cost United £60,000 after winning another title medal in 1960.
An ever-present in the successful 1964/65 league campaign, he scored 15 times and went on to make 112 starts for United before joining Blackburn Rovers for £40,000. He scored one of the goals in the famous 5-1 demolition of Benfica in Lisbon in 1966.
Connelly was also a member of England's World Cup-winning squad of 1966, one of three Reds representatives in Sir Alf Ramsey's party, but had to wait until 2009 to get a medal as FIFA decided to change their rules of only those featuring in the final receiving the honour.
Looking back on his career with United Review, he said: "Burnley's success was essentially down to teamwork and relying on tactics, whereas there were great individuals at United. Bobby Charlton, the best player I ever appeared alongside, would shoot from everywhere and you could also rely on Denis Law and George Best producing pieces of brilliance to win us a game."
Speaking in 2010, he added: "I am enjoying retirement in Barrowford, Lancashire, with my wife Sandra in the house we’ve lived in since 1966. We have a motor home, and since retiring have travelled all over Great Britain and Europe. Sandra and I ran a fish and chip shop in Brierfield (near Burnley) for 31 years."