Nigeria international Odemwingie fired home from 25 yards to deny the Owls a moral-boosting victory following Nejc Pecnik's first-half opener.
Former Baggies goalkeeper Chris Kirkland saved comfortably from Zoltan Gera early on before Pecnik netted 11 minutes before the break.
The Slovenian midfielder collected the ball on the right and his cross looped off the head of Youssouf Mulumbu and over stranded Albion keeper Ben Foster.
Wednesday suffered a blow early in the second half when midfielder Chris Lines left the field on a stretcher following an awkward fall.
Steve Clarke's men searched for an equaliser and Hungarian Gera fired wastefully wide from 10 yards before Odemwingie found the bottom right-hand corner of Kirkland's goal moments from the end.