Adrian produced a stunning early save, touching Oscar's shot against the crossbar, but the busy night the visiting goalkeeper might have expected never truly materialised.
Chelsea dominated the play and Demba Ba was also denied by the woodwork late on, but the home side were persistently thwarted by a West Ham defence as tenacious as it was deep-lying.
Andy Carroll also provided a muscular outlet in attack on an evening only soured by events elsewhere for manager Sam Allardyce, as Sunderland's win over Stoke City left them two points from safety.
Chelsea are a further point shy of the Premier League summit after Manchester City claimed top spot with victory at Tottenham Hotspur.
John Obi Mikel replaced David Luiz in the Chelsea midfield – Jose Mourinho's only alteration to the XI that beat Manchester United 3-1. Carroll's inclusion was one of five changes made by West Ham boss Allardyce.
Carroll appeared eager to make up for time lost in an injury-ravaged campaign and was a muscular presence throughout the opening period. The former Liverpool forward had a penalty claim rejected when he tangled with Cesar Azpilicueta in the sixth minute.
Adrian made a magnificent fingertip save in the ninth minute, touching Oscar's arching strike on to the crossbar.
The visitors served warning of their set-piece threat when Petr Cech plunged to his right to keep out a header from the recalled James Tomkins and Kevin Nolan was unable to readjust and convert the rebound.
Tomkins was called into action at the other end in the 15th minute, successfully coaxing Samuel Eto'o into blazing over after Eden Hazard deftly nodded Branislav Ivanovic's searching ball backing into the former Barcelona striker's path.
West Ham frustrated their hosts admirably for the majority of the half, but Adrian was forced into a brace of sharp saves on the stroke of the interval from John Terry and Eto'o.
Carroll once again found referee Neil Swarbrick uninterested in a penalty shout when he appeared to be struck by Terry's flailing arm – an incident that punctuated sustained Chelsea pressure in the first 10 minutes of the second half.
Adrian and the massed defensive ranks before him held firm in the face of speculative efforts and when Ramires found space in the box he badly miscued his half-volley.
The Brazilian was outdone in the blooper stakes by Carroll's wretched 60th-minute air shot, but the attack buoyed West Ham, briefly sparking them as an attacking force.
Frank Lampard, who netted in Chelsea's 3-0 win over his former club in November, was introduced in place of Azpilicueta and almost had an immediate 64th-minute impact as Tomkins and Matt Taylor sprawled to block a trademark drive.
It was then Hazard's turn to flash wide and Lampard found himself crowded out by Taylor once more - Chelsea's evening descending into one of repetitive disappointment.
Frustratingly for Mourinho, his men waited until stoppage time to genuinely rally – Ba prodding against the post and Lampard drawing a stunning save either side of Eto'o slamming home amid confusion after a free-kick was awarded in the West Ham box.
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