The punk group said on Twitter they were “saddened beyond words” to share that Hunt had died on Monday afternoon in London.
Posting a black and white photo of Hunt holding a guitar case, they paid tribute by quoting the lyrics of their song Love You ‘Till The End, which was written by the bassist.
They quoted: “I know you want to hear me catch my breath / I love you till the end.”
In his own tribute, the band’s frontman, Shane MacGowan, described Hunt as a “great friend and a great bass player”.
“I am very very sorry that Darryl has passed on, he was a really nice guy and a great friend and a great bass player,” he wrote on Twitter.
“We will all miss him.
“May he have a happy state of eternal bliss and bless his family and friends, love Shane.”
Fellow bandmate Spider Stacy also paid tribute on Twitter, writing: “This is wretched. See you around the way, Daz. ‘It’s in the fridge’ #TrickyTrees”.
The musician and songwriter was born in Hampshire, England on May 4 1950.
Before he began playing bass in The Pogues in 1986, he was a part of the pub rock band Plummet Airlines and The Favourites.
He was involved in the Celtic-inspired band’s subsequent work including their 1988 album If I Should Fall From Grace With God which featured the Christmas hit, Fairytale Of New York.
Hunt featured in their material until their last album, Pogue Mahone, in 1996.
He has writing credits on a number of their songs including Love You Til the End.
The band consisted of MacGowan, Jem Finer, James Fearnley, Andrew Ranken, Terry Woods, Hunt and other musicians throughout the years.