The Hammers went into an early lead as Andy Carroll rose highest to meet a Mark Noble free-kick after only 5 minutes.
That goal was the only high point of the afternoon as Stoke came from behind to dominate the game.
The Potters equalised through Peter Crouch before half-time and then went ahead on 69 minutes through Arnuatovic. They then added a humiliating third from Odemwingie on 79 minutes on the counter-attack following a Hammers corner.