Hodgson's Albion troops completed the ultimate smash-and-grab raid to record their first win at Anfield for 45 years as Kenny Dalglish's Reds dominated but dropped yet more points at home.
Had the hosts converted all their chances they would probably have made it into double figures but their season-long goalscoring problems cost them dearly once again.
Baggies manager Hodgson, sacked by Liverpool in January last year after a disappointing six months in charge, will no doubt have taken great satisfaction on his first return to his former club.
Dalglish, the man who replaced him, will be wondering what his side have to do solve their problems in front of goal after twice hitting the woodwork and having a penalty claim rejected in addition to numerous other opportunities.
But having not broken West Brom's resistance there was always the risk of punishment and it was duly delivered by Odemwingie 15 minutes from time.
Liverpool have now won just one Premier League game at home in 2012 and scored only six goals in seven matches, and have dropped 27 points at Anfield in the current campaign.
With diving back on the agenda after recent incidents involving Manchester United's Ashley Young, the spotlight was turned on Maxi Rodriguez in the most contentious incident of the first half.
The Argentinian did appear to be caught by Billy Jones as he flicked the ball past him on the edge of the six-yard area but referee Neil Swarbrick waved away penalty appeals.
But Liverpool, without the injured Steven Gerrard, should not have had to rely on the official for a helping hand after a number of chances went begging.
Their previous nine draws and two defeats at home were a direct result of them failing to convert and the first half was almost a re-run of games gone by.
Players seemed to be queuing up to squander opportunities as dominance in possession failed to lead to an end product.
Dirk Kuyt, whose future at Anfield has been the subject of much speculation this week, set the tone in the 10th minute by dragging a shot wide when he had to at least hit the target.
Luis Suarez's cross-shot failed to either test goalkeeper Ben Foster or find a team-mate in the six-yard area and when he did have a shot parried by Foster, the rebound was blazed over by Rodriguez.
In between there was the penalty appeal before Carroll headed wide, Suarez nodded a looping effort on to the roof of the net and Daniel Agger had a close-range shot turned behind.
West Brom were restricted to two chances but had it not been for the brilliance of Jose Reina, returning after a three-match suspension, they would have been ahead.
The Spain international produced an instinctive one-handed save low to his left to stop Chris Brunt's 28th-minute volley before blocking Liam Ridgewell's shot when Graham Dorrans' corner dropped in the six-yard area.
If anything it got worse after half-time as the visitors' defence came under siege yet was never beaten, although that was often down to luck as much as design.
Jordan Henderson began the onslaught by smashing a shot against the crossbar with Foster well beaten, Suarez produced a trademark weaving run along the byline before firing over with Rodriguez also volleying into the Kop from another Carroll knockdown.
Kuyt continued the theme with a shot deflected onto a post by Ridgewell and Carroll both volleyed and headed over.
The Baggies' goal was leading a charmed life with Foster saving from Suarez, Carroll falling over as he tried to force home the rebound and Jay Spearing's shot blocked by Brunt on the line all in the same move before Henderson fired into the side-netting.
But the 75th minute brought the sucker punch which Reds fans who have watched similar games at Anfield this season were probably expecting.
Glen Johnson's attempts to wriggle out of a tight spot deep on the right of defence saw him give the ball away to Mulumbu, who played in Odemwingie to shoot past Reina.
It was only the Baggies' second goal in 12 visits to Anfield - the last coming when Garth Crooks scored in a 4-1 defeat in 1985 - but if Hodgson felt a degree of smugness he did well to hide it as he sat unmoved on the bench.
Still Liverpool's onslaught continued, with Jonas Olsson blocking a certain goal for Carroll and Foster claiming Suarez's narrow-angle shot at the second attempt, and at the final whistle an understandable air of disbelief wafted around Anfield.