The Tigers full-back left a cross-field pass way short to allow Odemwingie to pounce and go on to fire a powerful left-footed shot beyond Steve Harper on 62 minutes.
Stoke have now reached the magic 40-point mark courtesy of a run of only one defeat in nine games, while Hull remain on the outskirts of the relegation battle, seven points above the bottom three.
Below, Sports Mole analyses whether the result was reflective of the action in Staffordshire.
On target: 3
On target: 3
Was the result fair?
Hull may rightly feel aggrieved that they did not leave the Potteries with a point this afternoon. It was Stoke's Asmir Begovic who was the busier of the two goalkeepers and they were starting to take control of the match before gifting the hosts the only goal. They only have themselves to blame for the defeat.
This was one of the few days in recent weeks where Stoke's passing game let them down. It was particularly sloppy in minutes 15-45, which were hugely disappointing considering that they started quite brightly. They had much more joy when they played a bit more direct up to Peter Crouch, and Marko Arnautovic managed to get close to the ex-England frontman to good effect. The fact that they could only find the target with three of their 15 attempts does show that their finishing was often poor today, but that cannot be said of Odemwingie's well-taken goal, a very clean strike with his weaker foot. At the other end, Begovic made just as an important contribution with a great reflex save from David Meyler's deflected shot, and he also kept out Tom Huddlestone and Shane Long.
The Tigers played well for a couple of 20-minute spells, but were not able to sustain any period of prolonged pressure on the Stoke net, particularly after conceding the opening goal courtesy of Elmohamady's error. The link-up play between Long and Nikica Jelavic was again tidy, but the Croatian snatched at a couple of half chances, and another opportunity, arguably Hull's best, fell to the wrong man in the form of Liam Rosenior. They will be happy with how little their defence gave stand-in goalkeeper Steve Harper, deputising for the injured Allan McGregor, to do, and it was only really a blunder from one of their most consistent players this season which saw them opened up.
Sports Mole's man of the match
The Austrian, a summer signing from Werder Bremen, is improving with every game and is certainly a favourite with a crowd who are enjoying the more expansive Stoke that they are seeing this season. Arnautovic is a big part of that and, although the end product was not always there today, he was one of the few players to show the intent and willingness to try and create something in an otherwise very flat first hour. His link-up play with Crouch, who was also excellent, shows signs of a developing understanding.
Hull were just starting to find their feet in an attacking sense when Elmohamady threw their good work away by taking a completely unnecessary risk. His cross-field pass was way short of Rosenior, and Odemwingie pounced brilliantly to score in the decisive moment of the match.
It was a very quiet afternoon for the man in the middle Neil Swarbrick, who showed only two yellow cards throughout the match. Both were to Stoke players, one of whom, Odemwingie, was booked for an over-enthusiastic celebration. That's another daft rule that needs looking at, surely?
Stoke: The Potters may just have an eye on Newcastle in ninth place, but have a difficult task ahead of them next Saturday as they travel to Stamford Bridge, where Jose Mourinho's Chelsea are invincible.
Hull: They will be desperate to make home advantage count in the Premier League against Swansea City next week, so that they can focus on their FA Cup semi-final knowing that survival is all but secured.