The late singer of the Doors was convicted of exposing himself to a Miami concert audience in 1969.
But his surviving band mates have denied he did this, and fans' memories are mixed.
The Doors singer died in Paris in 1971, aged 27, before he could serve a jail sentence.
"Nobody would like to have that charge hanging over their head, even if they are dead," Doors guitarist Robby Krieger told the Associated Press news agency.
"I'm sure his family would be happy to see that go, especially since it never happened."
On the night of the March 1969 show in Miami, Morrison was so drunk that his band mates questioned whether he would be able to perform.
The singer was abusive to the audience, and he stopped in the middle of the show to issue a profanity-laced, anti-authority rant, then tossed a police officer's cap into the crowd. At one point, a live lamb was brought on stage.
Morrison then fiddled with his belt as if to expose himself, at which point recollections diverge.
Governor Charlie Crist and other members of the Florida executive clemency board, composed of members of the governor's cabinet, voted on the pardon at their last meeting before he leaves office next month.
Mr Crist is reported to have doubts about whether Morrison committed the crime, and has been considering a pardon since 2007 following urging by fans.
Keyboardist Ray Manzarek has long denied that Morrison flashed the audience.
"It never actually happened, it was mass hypnosis," he said.
"He was just doing a mind trip - as they would say - a mind trip on the audience and they totally fell for it."