Joe told Entertainment Tonight Canada that it was Monteith's mother who first discovered that her son had a drug problem.
"I guess he was involved a little more in drugs than anybody knew at the time. His mother who was there with him all the time, she started picking up on it, you know stuff was missing, money here, stuff like that."
Joe, who was estranged from his famous son until recently, said they could not believe the news when they first heard it.
Monteith's body was cremated in Vancouver following a private viewing with his mother, brother Shaun, and girlfriend Lea Michele.
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Joe did not attend the funeral and said he was "totally devastated" that he didn't get a chance to say goodbye.
"I always loved both my sons and am going to miss Cory very, very much. I'm saddened at the fact that I couldn't have been there to see my son before he was cremated."