The Hammers fell behind to an early Jonathan Walters goal as Stoke went in search of their first away league win in 15 outings but the Hammers ended up with a share of the spoils courtesy of a rare goal from Joey O'Brien.
Sam Allardyce's hosts struggled to create chances in the first half but were much more potent after the break and dominated proceedings following O'Brien's strike, only his second goal for the club.
The hosts saw more of the ball in the opening 10 minutes but Stoke showed their usual aerial threat on a couple of occasions, most notably when Ryan Shawcross angled a long ball forward which Peter Crouch knocked down into the path of the onrushing Charlie Adam, who could not keep his shot down.
Unsurprisingly Stoke's opening goal came from a set-piece, but it was not scored in the manner usually associated with Tony Pulis' side.
George McCartney cleared the ball behind after 12 minutes and Glenn Whelan rolled the corner across the ground and into the area where a clever winding run from the far past allowed Ireland team-mate Walters to convert with a thumping effort past Jussi Jaaskelainen.
West Ham were struggling to break down a resolute Stoke defence that had been given something to protect.
Stoke, playing on the counter-attack, pushed forward and almost doubled their lead through another corner.
Kevin Nolan came close to a sixth goal of the season as he finally forced Begovic into a decent save with West Ham starting to exert control.
The visitors came within inches of doubling their lead seven minutes before half-time as Steven N'Zonzi fired in a shot from 25 yards that crashed off the crossbar and bounced just in front of the goal line with Jaaskelainen well beaten.
West Ham's leveller came early in the second half as Mohamed Diame's shot was charged down and O'Neil delivered from the right for O'Brien to convert from the middle of the goal.
The hosts continued to press but couldn't find a winner as Stoke held out for a draw.