Stoke's position looks perilous after they failed to pick up a win in their last 10 matches, but their manager believes they are still in the scrap to stay up this season.
It was a cagey opening in the first 45 minutes, with former Potter Marko Arnautovic forcing a point-blank save from Jack Butland in what was an otherwise dour first half.
Shortly after Moritz Bauer produced a brilliant run and cross for Stoke striker Mame Biram Diouf, who could only scuff the ball over the cross bar. Arnautovic had strayed offside before directing in a 55th-minute header and the same player was flagged again when Edimilson Fernandes fired in a shot from 20 yards.
Diouf then missed Stoke's clearest chance, turning a Bauer pullback over the crossbar with a first-time strike that wasted a risky break.
West Ham were building pressure on the Stoke goal and Butland parried away a Cresswell free-kick to protect his clean sheet, Lambert turning to Crouch in search of a key goal.
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West Ham ace Carroll reflected on the last-gasp handball that put paid to a potential victor from Javier Hernandez.
Those vital four points have moved the Hammers six points above the drop zone, and a win on Monday night will move them up to 12th in the table and nearly guarantee their Premier League status for next season.
- On home soil, West Ham have lost just one of their last seven top-flight games (W3 D3), after losing three of the six before that (W1 D2).
David Moyes's team also had three second-half goals disallowed and it was a dramatic finale to a game that had taken an age to ignite.
But substitute Carroll equalised with a well-taken strike just four minutes after his introduction to dent the visitors' survival fight.
All the same, it was easy to see why they were averaging under a goal per game this season.