Odemwingie replaced two-goal hero Romelu Lukaku who took his goal tally to 12 goals - all in the Barclays Premier League - this season.
The 31-year-old received a mixed reception with boos intermingling with the applause that greeted Lukaku's exit from the action.
Clarke, when asked about the response of the Baggies fans to the Nigeria international, said: "I never noticed it as I was too nervous about throwing away a two-goal lead.
"But with his first touch he had a shot that went just wide. He got good applause for that, and I think the supporters would have been happy to see that go in.
"The Odemwingie thing meant I could never affect what the supporters are going to do.
"He's part of my squad and we're going to use him.
"He was always going to come back in at some stage. For me it's finished, the incident happened but that seems like a long time ago now.
"We have to move on and so should everyone else."
Clarke hailed another man-of-the-match performance from Lukaku who is on a season-long loan from Chelsea.
He scored the opening goal from the penalty spot and charged down a clearance from Sunderland goalkeeper Simon Mignolet for his second.
Clarke said: "Lukaku is in a really rich vein of form. You could see that with his second goal, when you're playing that well you get a little bit of luck.
"Physically, he's a great specimen, but he has quick feet for a big man.
"It's been good for us, good for Rom, and probably good for Chelsea as well."
Lukaku insists he can cope with the pressure of being an £18million player - his fee when he joined Chelsea from Anderlecht.
He said: "I don't feel pressure. People ask 'why did he cost that much' but that's just football.
"Wayne Rooney got bought for more than me and in the first year at Manchester United, he didn't have an easy year.
"I am reading his book at the moment. I am a very big fan of his book.
"It is very good for a player my age because he went to a big club at the same age as me and I can learn from that."
Sunderland could only manage a late consolation from Stephane Sessegnon but manager Martin O'Neill felt his side should have been awarded a penalty for handball by Baggies defender Gareth McAuley.
He said: "I felt we should have got something out of the game. That's the last three matches we've played where we shouldn't have been beaten.
"The big decision was the penalty kick which was given against us.
"But then there was a similar situation at the other end where it wasn't given. I've seen both of them back. Both are penalties.
"Their penalty gave them big lift. They got the help of a bizarre second goal. But we were never out of the game.
"We should've scored two or three times before West Brom's second goal."
O'Neill acknowledged Lukaku's contribution.
He said: "Lukaku was very strong and he looks like he has a good future ahead of him. He was a handful."