Albion went in behind at the break after conceding a soft 15th-minute goal to Cameron Jerome from a set-piece.
But it proved to be a game of two halves as Albion stepped up a gear in the second period and put City to the sword with two Odemwingie strikes and a Jonas Olsson header.
"I'm getting a lot of support from my team-mates and fans and that gives me confidence, which I think is translated into performances," the Nigeria striker told the WBA website.
"What I've noticed so far from my first four games is that West Brom play better in the second half. I think we try to be careful in the first half, to play safe and then go for goal.
"But today we conceded a goal and then started to doubt ourselves in the first half. The coach shouted a lot in the dressing room at half-time and told us to concentrate more and play our game.
"Birmingham had some good quality players in the midfield but I think the players we put in the middle helped us a little bit. Certainly, the second half was much better and a quick goal helped us to believe and go forward and get three points.
"We showed a lot of character today."