The rock musician had forgotten about the work, which was being kept in storage, until it was found "rolled up in a tube" in a locker along with a collection of 1970s stage props.
'Little Electric Chair' - a silkscreen from Warhol's Death and Disaster series - was presented to Cooper as a gift by his late girlfriend Cindy Lang, after they were introduced to the artist and spent time with him in New York's famed Studio 54.
A similar version of the artwork was sold in 2014 at Christie's for $10.5 million (£8m).
Cooper's long-time manager Shep Gordon said: "It was a rock 'n' roll time, none of us thought about anything."
"He [Cooper] was a very heavy drinker in those days," he told the BBC, explaining that Cooper's career was "like a rocket ship taking off back in the early 70s - he was working 100 shows a year".
Gordon said that part of Cooper's live show at the time involved him "getting electrocuted" in a prop electric chair that looked "very much" like the one in the painting. The prop chair was found among the items in storage, alongside the painting.
Cooper later entered rehab to deal with his drinking problem and "never really moved into his apartment in New York". The painting was consequently forgotten.
Years later, Gordon was speaking with an art dealer friend when talk turned to a Warhol which had recently sold for a large sum.
"I got hold of Alice and I said: 'Do you still have that Warhol?' And he said: 'I don't think so.'"
But they managed to track it down months later at the storage facility.
Speaking to the Guardian, Gordon added that Cooper thought he remembered having a conversation with Warhol about the picture, which he "thinks was real, but he couldn't put his hand on a Bible and say that it was".
The work has been confirmed as real by Warhol expert Richard Polsky - it is now believed that the artwork could be worth millions.
"You should have seen Alice's face when Richard Polsky's estimate came in," Gordon said. "His jaw dropped and he looked at me: 'Are you serious? I own that!"
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies