The Nigerian forward had not played for the Baggies since trying to force through a move to Queens Park Rangers on transfer deadline day.
Odemwingie was initially disciplined by the club and banned from first-team training, but Clarke brought him off the bench against the Black Cats and has insisted that the striker is "part of the squad".
Clarke said: "He is part of our squad and part of our team. The reaction was a bit mixed.
"He had a chance at the end and I'm sure the supporters would have liked us to have gone 3-1 up."
Odemwingie has scored five goals in 20 appearances for West Brom this season.