Alessandro Diamanti equalised Chung-Yong Lee's opener before the match turned in the 77th minute when Green spilled Gary Cahill's shot in front of Ivan Klasnic, who snapped up the gift to score from point-blank range.
Despite Green's clanger, Clarke backed him and the Hammers to bounce back in Sunday's derby with Chelsea at Upton Park.
"It's a great game for us," he said. "Nobody expects us to get anything and they have to go there and work as hard as they can. Football's all about mistakes and Robert will deal with it and he will recover. The best thing he can do is to go out against Chelsea and have the game of his life. He has to respond and I'm sure he will."
Clarke also denied that the boardroom upheaval at Upton Park is affecting the team.
West Ham are set to turn down a £46million takeover bid from David Gold and David Sullivan but unless the club get new investment then Zola will have to sell before he can strengthen the squad next month.
"Obviously we're aware it's going on but there's nothing we can do about it," said Clarke. "Inside the dressing room it's not spoken about."
