Bramble had a solid afternoon but was frustrated that Odemwingie managed to expose a big gap in the Wearsiders' defence for the all-important goal.
"If we're going to get into the top half of the table this season then we really need to be getting three points from places like West Brom," said the former Wigan defender.
"It was a really disappointing day for a lot of reasons. We didn't defend well for the goal and generally didn't play well as a team. It's especially disappointing to have only one point from the first two games.
"In the first game against Birmingham we flew out of the blocks and showed our potential.
"One of the toughest things to get in football is consistency, but we need to gel quickly because you can lose ground very quickly in the Premier League and it's not easy to catch up."