Wanderers agreed an undisclosed fee with the Tykes on Thursday and 24 hours later he agreed to personal terms on a three-year contract with the Black Country club.
Speaking at Thursday's press conference, Barnsley boss David Flitcroft said: "He wants to go and play for Wolves, that's the bottom line.
"It doesn't matter how you try and keep a player if his heart's not in it and his head's not in it. You're pushing water up a hill. I feel the last two week his head and heart's not been with us, but before that his commitment to the club's been impeccable.
"He's one of the most professional players we've had the pleasure of dealing with. He goes with our blessing, I just wish the last two games he'd have produced the football I know he's capable of."
Golbourne joined Barnsley from Exeter just over 18 months ago and boasts a wealth of experience having also played for the likes of Bristol City and Reading, while enjoying loan spells with Wycombe, Bournemouth and Oldham.
Golbourne has started all six of Barnsley's league and cup matches this term.