The Nigerian international capitalised on a poor header from Richie de Laet to turn the ball past young Norwich goalkeeper Declan Rudd.
Graham Dorrans went close to extending the lead for the Baggies just before the break, but his long range effort struck the post.
Odemwingie had a great chance to make the points safe for West Brom, but his 75th minute penalty kick, after Steve Reid had been felled, was parried away by Rudd.
Substitute Steve Morison had Norwich's best opportunity to level proceedings but his close range header flashed wide of goal.
Next up for Norwich is a trip to Bolton Wanderers, while Roy Hodgson will take his West Brom side to Swansea City. Both matches will be played next Saturday.