"Robin brought so much joy into my life and I will carry that joy with me forever. He was such a beautiful man. I was lucky to know him and I will never, ever forget him," Damon said.
"I truly hope the people in the media can find it within themselves to give his family some privacy during this horrible time."
'Good Will Hunting' star Ben Affleck also gave an equally moving tribute to Williams, who played a mentor to the Damon's onscreen character.
"What do you owe a guy who does that? Everything. May you find peace my friend," Affleck wrote on his Facebook page.
'Good Will Hunting' won critical acclaim, landing nine Academy Award nominations and earning Affleck and Damon the Best Original Screenplay Oscar, while Williams walked off with the Best Supporting Actor trophy for his role.