Diamond, 70, is well known for his song "Sweet Caroline," inspired by Caroline Kennedy and now used as the Boston Red Sox anthem.
Cooper (real name: Vincent Furnier) is known for his band's hard rock hits "Eighteen" and "School's Out."
"We've always been a hard-rock band. We just wanted to decorate it a little differently," he said during Monday night's ceremony at the Waldorf-Astoria hotel in Manhattan, wearing a blood-splattered shirt and a boa constrictor.
Waits, who has written songs for Bruce Springsteen, Rod Stewart, Johnny Cash and the Ramones, told the audience: "They say that I have no hits and that I'm difficult to work with ... and they say that like it's a bad thing."
Love has lent her voice to several of singer Phil Spector's hits including "Da Doo Ron Ron" and "Christmas (Baby Please Come Home)." She was inducted by Bette Midler who said, "Listening to her songs, you had to dance, you had to move, you had to keep looking for that rebel boy."
John Legend inducted Dr. John, a global ambassador for New Orleans, and remarked, "He has never stopped flying the flag of funk."
Russell is known for his hits "Delta Lady" and "A Song for You," and in recent years has collaborated with Elton John, who inducted him.
A tape of the ceremony will air Sunday at 9/8c on Fuse.