The England star blasted home with 14 minutes remaining to finally end West Ham's resistance.
The visitors made a lively start with some strong challenges, notably on Frank Lampard and Didier Drogba.
But it was Drogba who had the first effort on goal, when, on eight minutes, on the edge of the West Ham area, the Ivory Coast striker attempted a curling shot on goal, but the ball drifted wide of Robert Green's left upright.
For much of the first half, it was a case of tackle, followed by more tackles, with neither side obtaining dominance in possession or creating any goal-bound chances.
Danny Gabbidon was called into action to dispossess Drogba on 29 minutes after sublime set-up play by Joe Cole on the left as he fed Wayne Bridge. The left-back sent in the perfect cross, which was chested down by Drogba, but Gabbidon was alert enough to clear the danger.
Nolberto Solano almost scored a minute later when he broke clear and clipped a deft ball over Carlo Cudicini but the ball was too high and went narrowly over the crossbar.
Four bookings, all for fouls, two for each side, reflected the tense and aggressive nature of the game.
A late first-half rally by Chelsea almost saw Alex score via Lampard's corner, and a mix-up between Drogba and Juliano Belletti led to the ball moving across West Ham's six-yard box without a touch by either player.
Steve Sidwell had an excellent chance two minutes into the second half but Lucas Neill, who appeared to handle the ball, charged his drive down.
Sidwell, again, had a fine chance from Joe Cole's corner but headed over the West Ham crossbar.
Carlton Cole sliced the ball into Chelsea's side-netting on 58 minutes and a minute later Lampard drove a trademark shot just wide in an exciting see-sawing opening period.
A mistake by Belletti led to the ball falling to Luis Boa Morte on 61 minutes but the former Fulham striker's effort was saved by Cudicini.
A minute later Green produced two saves in quick succession to thwart the west Londoners.
The Hammers keeper then clattered John Terry after the defender had volleyed wide following Lampard's deflected 72nd minute drive.
Four minutes later the deadlock was finally broken. From an apparent offside position, Joe Cole collected a headed pass by Drogba, took the ball into the penalty area, rounded Green and lashed the ball high into the net.
Seconds later Gabbidon saved his side by clearing a fine Shaun Wright-Phillips cross. Dean Ashton had the Hammers' best chance of the second half but his 86th minute free-kick was high and wide.
But Chelsea closed the game down and finished deserved winners, but only just.