Buzzcock's lead singer Pete Shelley died of a heart attack at the age of 63.
The punk band is best known for their hit Ever Fallen in Love (with someone you should not have).
Their management told the BBC that Shelley had died on Thursday in Estonia, where he lived.
BBC music correspondent Lizo Mzimba said Buzzcocks, formed in Bolton in the 1970s, was considered as polished but musically no less influential than the Sex Pistols.
The band has tweeted that Shelley was "one of the UK's most influential and prolific songwriters and co-founder of the groundbreaking original punk band Buzzcocks."
His music has inspired generations of musicians over five decades with his band and as a solo artist, they said.
- Listen: Pete Shelley remembers the 70s
Charlatan singer Tim Burgess said Shelley's "perfect three-minute pop songs" was "the teenage soundtrack."
Teenage fan club vocalist Norman Blake tweeted how important Shelley's songs were in his youth and remain so today.
And singer Tracey Thorn, formerly of Everything But The Girl, said Shelley was a "fantastic songwriter."
The author Neil Gaiman tweeted "a part of my youth dies with him".
Music critic Stephen Thomas Erlewine said Buzzcocks were one of the most influential bands to have emerged in the first wave of punk rock, echoing their music in each of Hüsker Dü to Nirvana.
"The Buzzcocks were inspired by the energy of the Sex Pistols, but they did not duplicate the angry political attitude of the Pistols," he wrote in AllMusic.
"Instead, they brought that intense, brilliant energy to the three-minute pop song, and Shelley's alternating funny and embarrassing lyrics about youth and love were some of the best and brightest of his time."