Chevron joined the Pogues in 1984 and became a core member as the group made its name internationally in the 1980s and early 1990s with albums including "Rum, Sodomy and The Lash" and "If I Should Fall From Grace With God".
Chevron, whose real name was Philip Ryan, wrote the band's popular ballad "Thousands Are Sailing".
"He was unique. We'll miss him terribly. Dublin town, and the world, just got smaller," the Pogues said in a statement on their website. "His loved ones are in our thoughts."
Chevron, who was born in Dublin, started out with the Irish act The Radiators From Space, which has been described as Ireland's first punk band, but he moved to London where he joined the Pogues fronted by fellow Irishman Shane MacGowan.
As well as playing guitar for the Pogues, he turned his hand to banjo and mandolin and occasionally took lead vocals.
Chevron stayed with the Pogues after MacGowan left in 1991 and was replaced by Joe Strummer, former frontman of The Clash, but he quit three years later due to poor health spurred by drug and alcohol abuse. The band split up in 1996.
He later reformed the Radiators with ex-Pogues bassist Cait O'Riordan. When the Pogues reformed in 2001 and interest in their music revived, Chevron remastered the band's back catalogue on CD and took a big role in their annual reunion tours.
He was diagnosed with head and neck cancer in 2007 and was given a clean bill of health in April 2012 but the cancer returned. His last appearance was at a testimonial concert in his honor in Dublin two months ago.