Odemwingie, whose transfer from Lokomotiv Moscow had only been confirmed 24 hours earlier, wasted little time in making a name for himself at his new club, taking on James Morrison's pass and slotting home the winner with nine minutes to go.
The victory marked a spirited response from the Baggies, who had started the day bottom of the Premier League table after they were thrashed 6-0 at Chelsea last week.
Having received international clearance to play, Nigeria forward Odemwingie was straight in the thick of the action, latching onto Chris Brunt's measured through-ball in the eighth minute and shooting narrowly wide.
Albion continued to press forward, with Morrison chesting down a failed clearance and sending a volley wide of goal.
At the other end Darren Bent, wearing the captain's armband in the suspended Lee Cattermole's absence, looked to have beaten the offside trap, only for the referee's assistant to raise his flag as the England striker fired at Scott Carson.
Sunderland then won a free-kick just outside the box, but former Baggies man Kieran Richardson could only curl it over the bar.
Albion responded with Morrison breaking down the right, beating two men and delivering a cross that was just too high for Brunt.
The hosts' passing game was starting to come together and Jerome Thomas' cross just before the half-hour mark found Brunt, who couldn't get a shot away but teed up a rather wild effort from Marek Cech.
Soon after, Albion fans cried out for a penalty when they felt Jordan Henderson had handled inside the box, but referee Kevin Friend waved play on.
The Baggies kept up the tempo and Thomas almost made the most of an error of judgment by Simon Mignolet by rounding the Black Cats keeper near the byline, but there was no-one in the middle for him to pass to.
Cech then sent in a cross that Brunt met with an acrobatic but off-target attempt.
It was Sunderland who started the second half brighter, with Fraizer Campbell doing well to keep the ball in and hoisting a cross into the box which caused a scramble before Cech managed to clear.
Bent then got his head to Henderson's free-kick, but could not guide his effort the right side of the post.
Albion wrestled back possession and won a free-kick in a decent position, only for Graham Dorrans to put it high over the bar.
Giles Barnes, brought on at half-time for Thomas, then produced a useful cross, but Mignolet was able to dive on it.
Richardson was next to try his luck from a set-piece in the danger zone and, although going closer than Dorrans, he also missed.
Brunt then rose to head Barnes' cross goalwards, but Mignolet made a comfortable catch.
With a little over 20 minutes remaining, Sunderland boss Steve Bruce threw on Danny Welbeck and Boudewijn Zenden as the visitors went in search of a winner.
It was Albion who were back on the attack, though, with Barnes again the provider of a cross into the box which Mignolet just managed to divert away from a crowd of players.
Odemwingie then created some space for himself and attempted to unleash a piledriver on the edge of the box that powered straight into Paulo Da Silva.
Play switched once more and Richardson saw a curling shot loop over the bar before Nedum Onuoha headed Zenden's cross at Carson.
But it was the hosts who finally made the breakthrough in the 81st minute, with Odemwingie bagging his debut goal after collecting the ball from Morrison, turning and sliding a low shot under Mignolet.
Bruce reacted by putting on Martyn Waghorn for Cristian Riveros and the young striker nearly made an immediate impact but planted his effort wide from a corner.
West Brom were then inches away from making it 2-0 with Brunt cracking a long-range attempt against the crossbar, before Friend blew his whistle for full-time to ensure the Baggies secured the spoils.