West Brom took the lead through Odemwingie before Steven Fletcher gave the home side an equalizer.
And a strike from Jonas Olsson and substitute Keith Andrews and Odemwingie's second sealed the points, before the Nigerian rounded out the scoring in the closing stages.
Manchester City returned to the top of the Premier League table thanks to a Joleon Lescott goal as they defeated Aston Villa 1-0 at Villa Park.
Lescott was left unmarked in the six-yard box to turn the ball home after Gareth Barry had flicked back James Milner's corner
