The enigmatic Frenchman helped himself to a first-half double to take his goal tally for the season to 13 - with seven of them coming in the last three league games.
The win, which saw Chelsea leapfrog back over Liverpool at the top of the table, was their tenth successive away victory. That is a Premier League record and matches the run Tottenham put together in the old First Division in 1959.
Chelsea, who were given the lead by a stunning strike by Jose Bosingwa, were simply too good for Albion as they took their goal tally from their last four away games to 13.
Albion are now in real danger of getting cut adrift at the foot of the table following a run of one point from six games.
Manager Tony Mowbray admitted before the game that he feared Albion could be torn apart by Chelsea and his fears were well founded.
Ironically, Albion made the best start of the two teams and they were made to pay a heavy price for not capitalising on their impressive opening spell as Chelsea hit them with three goals in the final 11 minutes of the first half.
Albion's Scotland international midfielder James Morrison fired just over the bar in the first minute before Ishmael Miller wasted a golden chance to break the deadlock. Picked out by a pin-point pass from Borja Valero in the 15th minute, Miller dragged his shot wide from 12 yards.
It was a turning point as Chelsea then went on to dominate affairs. Salomon Kalou went close from long range before Florent Malouda was twice denied by goalkeeper Scott Carson.
But, despite the best efforts of the England international, there was nothing that Carson could do when Bosingwa literally scored a goal out of nothing in the 34th minute.
The Portugal international right-back picked up a short pass from Malouda and took a couple of steps infield before curling home a 25-yard drive.
Carson got his hand to the ball but could only deflect it on to the bar before it fell into the net.
That was effectively game over and Chelsea doubled their advantage through a cool piece of finishing from Anelka. Released by a Malouda header, Anelka surged into the area before placing his shot past Carson at his near post.
Anelka was simply unstoppable at this stage and came up with another smart finish in stoppage time when he collected a cross-field pass from Kalou and again pounced at Carson's near post.
Chelsea went into cruise control in the second half and Anelka was unlucky not to complete his hat-trick when Jonas Olsson blocked his goal-bound drive in the 56th minute.
To their credit, Albion kept battling and substitute Filipe Teixeira was inches away from a consolation strike on the hour when his 25-yard drive dropped just over the bar while Do-Heon Kim flashed his angled shot just wide.