He died in New York from complications of rheumatoid arthritis, colitis and pneumonia, the band announced on Monday.
Frey and bandmate Don Henley became one of history’s most successful songwriting teams with such hits as Hotel California and Life In The Fast Lane.
The duo formed the Eagles in Los Angeles in the early 1970s, together with guitarist Bernie Leadon and bassist Randy Meisner.
A statement on the band’s website said: “Words can neither describe our sorrow, nor our love and respect for all that he has given to us, his family, the music community and millions of fans worldwide.”
Sir Elton John said he was “in shock” and told ITV’s Good Morning Britain: “I didn’t know he was sick.
“I found out on the way here that he died and it’s, you know, I don’t know what’s going on at the moment. It’s not good.”
Tributes poured in on social media, with One Direction star Niall Horan tweeting lyrics from the band’s single No More Cloudy Days.
He posted: “These cloudy days, make you wanna cry. It breaks your heart when someone leaves and you don’t know why.”
Comedian and actor Steve Martin wrote: “MT: Shocker. My friend from the early days, and important member of Eagles, has died. We loved you, Glenn Frey.”
Henley said crossing paths with Frey in 1970 “changed my life forever, and it eventually had an impact on the lives of millions of other people all over the planet”.
The Eagles Greatest Hits collection and Hotel California sold more than 20 million copies each and are among the best-selling albums of modern times. The band’s total album sales top 100 million.
The band broke up in 1980, with Frey and Henley also becoming estranged for years.
Henley had vowed the Eagles would reunite only when “hell freezes over”, which became the name of the 1994 album when they re-formed.
Frey’s health problems, including diverticulitis, dated to the 1980s. He blamed in part his years of “burgers and beer and blow and broads” and later became a fitness advocate.