Begovic was unable, however, to prevent Manuel Da Costa doubling West Ham's lead before the interval and Thomas Hitzlsperger rifling home a third goal late on.
The Potters are 12th in the Premier League and having appeared to be in a position of relative safety, a sixth consecutive league defeat away from home has left Pulis' side just four points above the relegation zone.
"We need to get to 40 points as quick as we can," added Pulis. "It's no good worrying about what other people are doing.
"You've got to be dogmatic, single-minded and focus on what you've got to achieve as a group.
"We've got to make sure that we're better defensively than we were on Saturday, especially through the middle of the pitch."
The Potters have the chance to gain revenge on Sunday when the two teams meet again at the Britannia Stadium in the last eight of the FA Cup.
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