Serbian striker Nikola Zigic dealt the fatal blow when he headed home a Seb Larsson free-kick midway through the second half to sink the sorry Hammers.
The big striker nodded home from a Sebastian Larsson free-kick to seal the win and move Blues out of the bottom three and up to 16th in the Premier League table.
Hammers substitute Demba Ba rattled the woodwork on his debut but otherwise, the hosts rarely threatened Blues' goal and manager Avram Grant's position will now undoubtedly come under more scrutiny as a result.
