The defender, who had been battling dementia, made 407 appearances and lifted two trophies for the Tannadice side over 10 years.
His wife Amanda has been campaigning for dementia and Alzheimer's awareness, and met with health secretary Alex Neil last month.
Alzheimer Scotland joined football clubs and fans in paying tribute to Mr Kopel.
Mr Kopel, who was inducted to the Dundee United Hall of Fame in 2011, started his career with Manchester United as a schoolboy in 1964.
He made 12 appearances for the Old Trafford side, including a European appearance against Anderlecht, before moving to Blackburn Rovers in 1969.
He was signed by United manager Jim McLean on New Year's Day 1972, and became a mainstay at the right-back position in a team which won back to back Scottish League Cup honours in 1979 and 1980, the club's first major trophies.
After leaving Tannadice to take up a coaching position with Arbroath in 1982, he returned to United on several occasions to help out with the reserves.
A spokesman for the club said: "Our thoughts are with all of his family and friends at this very sad time.
"An indication of the affection with which he was held by all Arabs is illustrated by the fact that a supporters' club was named in his honour and by the genuinely warm welcome he received at the many functions and events he attended over the years."
Alzheimer Scotland added: "Deeply saddened to hear of the death of Frank Kopel. Our sincere condolences to his family and friends."
Mr Kopel was diagnosed with a mixture of vascular dementia and Alzheimer's disease in 2009.
His wife Amanda mounted a campaign to raise awareness of the strain on carers looking after people with early-onset dementia, and addressed the Scottish Parliament's public petitions committee.
She has called for free personal care for dementia sufferers under 65, and health secretary Alex Neil pledged to bring her concerns to Parliament after meeting the couple at their Kirriemuir home last month.

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The 21-year-old, from Meon Valley Golf Club, followed in the footsteps of fellow Hampshire golfer Scott Gregory in 2016 by winning the trophy.
The victory guarantees Ellis will play at Royal Birkdale next month as well as the Masters and US Open next season.
"The magnitude of what I did is starting to sink in," Ellis said.
Ellis came from four holes down with five remaining in the 36-hole final to beat Australia's Dylan Perry at Royal St George's in Kent on Saturday.
Victory was only secured after two play-off holes for the man who became the youngest to win the English Amateur Championship, aged just 16, in 2012.
"I still feel a little numb," he told BBC South Today. "The task was getting a lot tougher to come away with the trophy and I knew I just had to keep myself going.
"It all came out when the winning putt was sunk. It was years of hard work and a lot of disappointment matched with the excitement of holing that putt."
Ellis has just completed his third year on a scholarship at Florida State University, where he hopes the experience will stand him in good stead for an appearance at Augusta next season.
He revealed US Open champion Brooks Koepka was among those to text their congratulations after Saturday's win.
"Justin Rose and Scott Gregory also sent some nice messages," Ellis said. "It's been really nice to receive those and I'm really excited for the opportunities ahead."
The Open begins at Royal Birkdale on 20 July.
Dundee United football legend Frank Kopel has died aged 65.
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Harry Ellis admits he still "feels numb and in shock" after securing his spot at The Open in July by winning the British Amateur Championship.