The home side took the lead 10 minutes before half-time when Odemwingie’s deflected shot crept past Southampton goalkeeper Paulo Gazzaniga.
The Saints came close just after the interval, with Jay Rodriguez’s cross finding Rickie Lambert, but the striker’s effort smashed against the crossbar.
West Brom wrapped up the win after Shane Long’s cross was headed past Gazzaniga by Odemwingie, leaving Southampton with eight defeats in their opening 10 Premier League games.
“To bounce back from the last two difficult games where we lost like we did was superb and I felt we deserved the win,” Foster told BBC Sport.
“We have conceded some unlucky goals recently so it is great to get back to winning ways and also keep a clean sheet. It is a cliche but we are not getting carried away, we will take each game as it comes.”
Meanwhile, match-winner Odemwingie believes West Brom need to keep being adventurous going forward after the Nigerian took his tally to three goals this term.
“This was a good response from a good team,” Odemwingie added.
“We needed a bit of luck and it is simple, if you keep shooting you will score, we are happy with the three points.”
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