Allardyce has come under fire after the London club's run of seven Premier League games without a win and humiliating FA Cup and League Cup thrashings against Nottingham Forest and Manchester City respectively in the last week.
But Carlton Cole scored shortly before half-time at the Cardiff City Stadium on Saturday, before Mark Noble sealed the points in second-half stoppage time to ensure Cardiff replace Allardyce's men in the bottom three on goal difference.
They were not to be denied three precious points, though, and Andy Carroll also made his first appearance of the season as a second-half substitute, the striker setting up Noble's goal as Ole Gunnar Solskjaer was beaten in his first home game in charge of Cardiff.
West Ham started brightly and Cardiff breathed a sigh of relief when Stewart Downing's fourth-minute cross from the right evaded everyone and struck the far post, before David Marshall denied Tomkins from the rebound.
Demel then became West Ham's latest injury victim when he was caught in the face by team-mate Roger Johnson's elbow after only seven minutes and was carried off wearing a neck brace following a lengthy stoppage.
Cardiff came tantalisingly close to opening the scoring after 31 minutes, when Kim Bo-Kyung’s shot deflected over Adrian and struck the crossbar before bouncing down on the line before Frazier Campbell fouled the goalkeeper following up.
Cardiff were almost caught out again a minute later, but Matt Taylor's right-foot effort whistled just past the far post.
Norway midfielder Eikrem came on to make his debut after 64 minutes and Bellamy curled a free-kick just wide six minutes later as the hosts' frustration grew.
However, the visitors stood firm and Noble sealed the points in stoppage time when he drilled the ball into the far corner after being played in by Carroll.
