Chris Brunt put the Baggies ahead and they doubled their advantage courtesy of a Green own goal, though the former West Ham goalkeeper appeared to be fouled by the Baggies striker.
Referee Chris Foy allowed the goal to stand, though the Hoops got back into through Djibril Cisse.
"I didn't think it was a foul," said the Scot, whose side remain in the hunt for at least a top-five place thanks to the win.
"I just saw the ball go up and then in the net. I was confused. I've seen it on the video and I don't think he has done a lot wrong.
"I didn't think it was much of a foul to be honest.
"And with the penalty appeal if it hadn't have hit his hand it would have hit his face so I think it's the correct decision."