Goals from Victor Moses and Mame Biram Diouf gave the Potters a 2-0 lead but the Hammers fought back to stay in the top four of the Premier League
Sam Allardyce's men arrived at the Britannia Stadium on Saturday fourth in the Premier League having won each of their previous three games, including a victory against champions Manchester City last weekend.
But the London club looked a shadow of that side without injured star striker Diafra Sakho and Victor Moses' first Premier League goal in 14 months gave Mark Hughes' side a deserved first-half lead before Mame Biram Diouf added a second after the break.
Enner Valencia pulled one back on the hour mark before setting up Stewart Downing's equaliser in the 73rd minute after Diouf had missed a great chance to seal the points for Stoke.
This sparked a frantic finale as both sides pushed for a winner, but neither could find the decisive touch.
Sakho's absence may have hampered West Ham, but the hosts also started the match without their chief attacking threat as Peter Crouch served a suspension, along with defender Phil Bardsley.
But Stoke's replacements combined to create the first opening on 10 minutes.
Geoff Cameron was allowed to burst forward from right-back time and again in the opening exchanges and almost made West Ham pay when he teed up Bojan 15 yards from goal. However, his shot was blocked bravely by James Collins.
Bojan drew confidence from his early opportunity and delivered a dangerous low cross from the right in the 22nd minute after evading Morgan Amalfitano, only for his centre to somehow escape the toe of a lunging Diouf.
Stoke were on top and gained the lead their dominance deserved in the 33rd minute.
It was not pretty, but Moses cared little, forcing the ball over the line from close range as it fell at his feet via Bojan and Diouf, after Carl Jenkinson failed to clear Cameron's cross.
Diouf could have doubled the advantage in first-half stoppage time, but was denied by a fantastic save by Adrian, after West Ham once again failed to cope with a ball into the box from Cameron.
However, there was nothing that Adrian could do to prevent Diouf from doubling Stoke's lead 11 minutes into the second half.
Aaron Cresswell was out-muscled by Jonathan Walters on the right and he then crossed for Diouf to head in from three yards.
The game looked to be getting away from the visitors until Alex Song won the ball with a robust challenge in midfield, releasing Downing. The winger's curling right-wing cross was headed in at the back post by Valencia to give West Ham hope with 30 minutes to play.
Diouf had a chance to put the game beyond the visitors 10 minutes later as he met Erik Pieters' perfect whipped free-kick, but he could not direct his effort on target.
And Diouf was made to rue that missed opportunity as Downing and Valencia combined once again to level. This time the Ecuadorian supplied the cross from the byline and Downing arrived in the box to smash home with his right foot.
West Ham's defence came under greater pressure in the closing stages as Stoke reverted to type with aerial balls into the box, but Adrian was not unduly tested.