The visitors, whose manager Avram Grant was observing Yom Kippur and not present at the game, took the lead in the first half thanks to Scott Parker's close-range finish.
Stoke came back strongly after the break, with Kenwyne Jones following up his goal against Aston Villa on Monday by heading home in the 48th minute after connecting with a cross from the impressive Jermaine Pennant.
The two teams had lost seven out of eight Premier League games between them before the start of play and it was a scrappy opening.
Rory Delap sent a long throw into the box which Abdoulaye Faye could not connect with properly and at the other end Carlton Cole struck an effort well wide of Thomas Sorensen's goal.
Victor Obinna had a little more conviction moments later when he fired in a volley, but it took a deflection and Sorensen was able to catch comfortably.
Robert Green, who had spilled a Didier Drogba free-kick in the build-up to a goal in the Hammers' previous match against Chelsea, almost repeated the error when he failed to deal with Pennant's delivery into the box, but his blushes were spared as Robert Huth hit the loose ball against the post.
Valon Behrami then picked up the game's first booking to give away another free-kick in a similar area which Pennant used to tee up Jones, but the Trinidadian sliced his shot over the bar.
Having seen Stoke fail to make the most of their set-pieces, West Ham then took the lead via a free-kick of their own.
Obinna crossed it into the box in the 32nd minute and the ball bounced off two red and white shirts before Parker prodded it over the line.
Looking energised, the visitors almost doubled their lead within two minutes as Frederic Piquionne unleashed a drive from the edge of the area which cracked against the bar.
Stoke attempted to regroup and Jones got his head to another Delap throw-in but sent it over.
Tony Pulis' half-time team-talk had inspired the Potters to come from behind and win against Villa and it seemed as if the manager's words at the interval had made an impact again as the hosts pulled level within three minutes of the restart.
Jon Walters released Pennant down the right channel and the former Arsenal and Liverpool winger played in a delightful cross which Jones headed in at the far post.
Former Hammer Matthew Etherington had a chance to put Stoke ahead soon after having been fed by Jones, but saw his effort deflect behind for a corner.
Pennant was providing plenty of impetus and delivered another great ball into the danger zone which Walters was just unable to meet.
Green then gave a reminder of his quality by pulling off a fantastic save, tipping Jones' strike against the post after the striker had outmuscled Matthew Upson.
Pulis brought on Ricardo Fuller for Walters with 20 minutes remaining and Pennant lofted another handy ball into the box for Jones, who was flagged offside.
West Ham had barely mustered anything in an attacking sense since half-time, but suddenly called Sorensen into action, the Danish goalkeeper diving at full stretch to palm Obinna's shot around the post.
From the resulting corner Mark Noble sent in a cross which Manuel Da Costa headed goalwards, but again Sorensen kept West Ham at bay, getting down low to his left.
Jones was then substituted, with Eidur Gudjohnsen receiving a fantastic reception from the Stoke fans as he came on for his debut.
With 10 minutes to go Fuller came close to finding a winner with a header from Delap's throw which bounced into the turf and up off the bar.
Green then pushed a strike from Etherington over as West Ham hung on to secure a point.