The Potters' first-half dominance was reflected in centre-back Robert Huth's eighth goal of the season but Frederic Piquionne's clear handball was missed by referee Mike Jones as the striker went on to score.
When former Hammer Matt Etherington missed a penalty seconds after the interval it seemed it was not to be Stoke's day but Danny Higginbotham drilled a free-kick through the wall and goalkeeper Robert Green could only push it onto a post and into the net.
Forward thinking: Pulis (left) looks to lift his side whilst Hammers boss Grant watches on in dissaray
Pulis was grateful to be able to sweep aside any criticism of the match officials.
'We have been very unfortunate this year, we have copped quite a few (bad decisions) in the league.
'We were very good for 25 minutes and then when they get their first shot on goal from that it was disappointing.
'From where I was standing it looked a stone-banker handball.
'Matty came out in the second half and missed a penalty and that knocks you down again but great credit to the players - for a lesser team that would have affected them.
'For two and a half years at this club, since we were promoted, we have been written off.
'We have got to three quarter-finals of cup competitions now we have a semi-final.
'We've finished 12th and 11th in the Premier League and the players and chairman deserve a hell of a lot of credit.'
Head start: Robert Huth gives Stoke the lead after rising to meet Rory Delap's dangerous throw-in
Stoke, who are into the last four for only the fourth time in their history, now face Bolton at Wembley
'I think it is being disrespectful to Bolton to say it is the draw we wanted, we just wanted to win today.
'We will treat them with total respect. If you look at Bolton and their history, they have been a Premier League side a lot longer than us and have established Premier League players.
'We will respect them and we know it will be a tough game for us,' added Pulis.
West Ham boss Avram Grant was disappointed with some of the decisions his side encountered in the second half, claiming James Tomkins should have had a penalty late on instead of being penalised for a foul on Jon Walters.
Over the line: Stoke pair Jon Walters and Ryan Shawcross celebrate after West Ham keeper Rob Green fails to keep out Higginbotham's strike
He said he felt the referee tried to even things up after missing Piquionne's handball.
'I think maybe this is the reason the referee started the second half very strange.
'Until they scored [the second] goal he gave fouls, a penalty, everything for them.
'It was a penalty for us at the end which he didn't give. What happened with Tomkins was more bushido (the Japanese code of conduct for samurai warriors).
'Maybe he felt a little bit guilty. I think it was the effect of the first goal for us,' said the Hammers boss.
Winner: Higginbotham celebrates at the final whistle
Meanwhile, Stoke matchwinner Danny Higginbotham is relishing a trip to Wembley with the Potters set to face Bolton as a reward for their win against West Ham
Higginbotham blasted in a powerful second-half free-kick to settle the quarter-final tie which took the Potters into the last four of the FA Cup for the first time in 39 years.
He told ITV Sport: 'It's a great feeling to have got to Wembley, we're delighted and we're just going to enjoy today, with a tough semi-final against Bolton ahead.
'West Ham are a great team, they showed that last week against us (when they beat Stoke 3-0 in the Premier League) but we thoroughly deserved it - we got stuck in. We got the lead and the second time we got it we were able to stay ahead.
'We've got great belief within the squad. We've had a disappointing run of games recently but we know we've got the character in there to come out of it.
'It's a great club and it's come on in leaps and bounds in the last few years.'
Robert Huth gave Stoke the lead but the Hammers equalised through Frederic Piquionne, although referee Mike Jones missed a handball in the build-up.
Matthew Etherington then missed a penalty - controversially awarded by Jones for a foul by Scott Parker on Etherington - early in the second half, before Higginbotham blasted home the decisive goal.
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?Explore more:People: Matthew Etherington, James Tomkins, Tony Pulis, Robert Green, Danny Higginbotham, Robert Huth
Source: Daily Mail
Source: Daily Mail