The Nigeria striker capitalised on an inexcusable, underhit pass by Glen Johnson to fire past Pepe Reina, who had made two top saves earlier but stood static as the ball seemed to fly inches past him.
The goal came in a rare excursion past halfway for the Baggies, who rode their luck after sustaining a battering from Liverpool, who hit the woodwork twice, forced a series of saves from Ben Foster and missed countless good chances.
The result - West Brom’s first victory at Anfield in 45 years to the day - means Liverpool stay eighth, just ahead of Fulham on goal difference, with the Baggies a point behind in 10th.
After a friendly reception from home fans for Hodgson on his first return to the club that fired him last season, the first half remained goalless, although not for want of trying.
Liverpool dominated possession and created multiple opportunities, denied by a combination of weak finishing and last-ditch defending.
West Brom, meanwhile, had two great opportunities of their own but Reina pulled out marvellous stops to deny Chris Brunt and Liam Ridgewell. He would undo that good work later.
That was pretty much it as far as attacking endeavour went for the visitors. The Reds were by far the ascendant side: with the inventive Luis Suarez pulling the strings, Dirk Kuyt put wide, Daniel Agger failed to convert from close range, while the Norway defender’s low finish was kept out by Ben Foster’s foot.
Suarez missed some chances of his own too, taking too long to finish and allowing Ridgewell to block, heading over when found unmarked by Jose Enrique, and forcing a fine stop from the busy Foster, with Maxi Rodriguez blasting the rebound over.
Maxi had earlier been the centre of controversy when referee Neil Swarbrick pointed for a goal kick after the Argentine went down in the box, replays suggesting he may have exaggerated contact from Billy Jones.
Jones, Jonas Olsson and Gareth McAuley put in countless blocks too, the best coming when Olsson denied Suarez a finish after a fine counter-attack down the Reds right.
But the hosts could not break the deadlock, with some boos ringing out at the break.
Undeterred, after half-time Liverpool picked up where they left off as a lovely chest and flick by Suarez again saw a stretching Agger fail to finish from close range, while Jordan Henderson hit the bar with a rasping drive.
Liverpool were carving the Baggies apart, but they could not convert - the story of a season that has so far seen them score fewer goals than Blackburn Rovers.
The build-up play was, at times, excellent, but the finishing weak, encapsulated when Suarez went on a fantastic mazy run before blasting high and wide from 10 yards.
Liverpool hit the post again as Kuyt latched on to a Suarez through ball, his low angled drive deflected by Ridgewell, coming back off the woodwork but behind the arriving Suarez - the 30th time the Reds have found the woodwork this season, more than any other Premier League side.
Their profligacy was not just down to bad luck - Andy Carroll, employed for his aerial threat, kept mistiming his jumps, heading over from great positions on two occasions after fine crosses from Kuyt and Johnson.
They were all over West Brom though, with Foster getting muddy as he pulled off a string of saves, some fine - such as the double stop from Suarez and Carroll, with Brunt clearing Jay Spearing’s follow-up off the line - and some sloppy, such as when he spilled an Enrique cross-shot.
The Baggies could barely pass halfway, although when they did Shane Long really should have scored, dithering and holding the ball up when played clean through by a long ball, possibly unaware of his options.
Craig Bellamy entered the fray, given a brief to run at Ridegwell, who struggled when confronted with pace.
But he had a limited impact, although was involved as Henderson hit a low drive wide but, for all their chances, you just fancied Liverpool to press the self-destruct button.
They did so in style when, with a quarter of an hour remaining, Johnson initially did well to dispossess Youssuf Mulumbu: he then hit a sloppy pass across the face of his own goal, so poor and ill-judged that it merely returned possession to Mulumbu, who found Odemwingie. His finish was not the greatest either - it was about a foot to the left of Reina - but the Spanish keeper just stood transfixed as the ball spun past him and in.
Liverpool tried to rally but the replacement of Spearing with identikit midfielder Jonjo Shelvey was hardly an inspiring change.
Foster did his best to help them out though, spilling a pair of daisy-cutters but saved first by Olsson’s block on Carroll and then by his own torso.
Five minutes plus of injury time barely helped Liverpool, who looked beaten mentally, as the Baggies moved into the top half.