The Nigerian striker scored the winning goal as Steve Clarke’s side saw off the European champions at the Hawthorns and climbed to fourth in the table.
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Odemwingie netted from a Shane Long cross, while the Irishman had also grabbed the game's opening goal in the 10th minute.
The Irish striker's grandmother died on Friday evening, and the former Lille man hailed his strike partner's determination to play, despite the bereavement.
"I wondered why he was a bit quiet," the 31-year-old admitted. "After the match we were celebrating with some music in the dressing room and he wasn’t with us. He deserves all the respect for playing."
The victory leaves West Brom five points off league leaders Manchester City, three ahead of fifth-placed Everton.