Luiz Felipe Scolari's side equalled Tottenham's 48-year-old record with a 10th successive away win in the top flight and surpassed its own previous best Premier League run away from Stamford Bridge set four years ago.
Anelka's double took his tally to 12 league goals for the season and eight in the last four games after Jose Bosingwa had broken the deadlock.
Chelsea's comfortable success, courtesy of three goals in 12 minutes before half-time, saw it overtake Liverpool on goal difference at the top of the table.
The Blues have now scored 19 goals in seven away league matches this season and conceded just one.
Albion showed plenty of promise in the opening half hour and knocked the ball around in the style demanded by manager Tony Mowbray.
But it lacks a cutting edge and is not strong enough defensively and remains anchored at the bottom of the table after collecting just one point from its last six games.
Albion began well and James Morrison was only just too high with a powerful 20-yard effort.
The first decent chance fell to West Brom after a powerful 50-yard run by Borja Valero. He reached the edge of the box before picking out the run of Ishmael Miller but the former Manchester City striker dragged his shot across the face of goal.
Chelsea retaliated and Salomon Kalou drilled a rising shot past the post from a wide angle. The visitors started to find more space near the Albion goal and the home side was indebted to Scott Carson for keeping it on level terms.
Malouda's first time ball was perfectly weighted for the run of Lampard into the inside left channel and his powerful drive was beat out by the England goalkeeper.
Carson was tested for the second time in turning away a swerving drive from Malouda after he had advanced towards the box unchallenged.
But out of nothing a superb strike from Bosingwa gave the Blues the lead after 34 minutes.
There appeared to be little danger but Bosingwa suddenly unleashed a powerful left-footed drive from 30 yards out which Carson got a hand to but could not prevent from crossing the line.
Albion was stunned and it needed a fine tackle from Jonas Olsson to deny Malouda as he lined up a shot inside the area.
But after 38 minutes Anelka doubled Chelsea's lead, seizing Malouda's headed pass and racing into the area to beat an exposed Carson.
Chelsea now threatened every time t ventured forward and in first-half injury-time Anelka struck again, beating Carson from a narrow angle with Kalou this time the provider.
Mowbray made a half-time substitution, taking off striker Roman Bednar and replacing him with midfielder Kim Do-Heon in what seemed a damage limitation exercise.
Chelsea was soon back on the offensive and Carson managed to block an effort from a narrow angle by Malouda.
Anelka had the chance to go for his hat-trick when in space 15 yards out but he unselfishly attempted to tee up Deco, who was denied by a last-gasp tackle from Paul Robinson.
The Chelsea players were now queueing up to take shots at goal and Abdoulaye Meite threw himself in front of a shot from Anelka.
Bosingwa picked up the first yellow cards of the game for challenges on Miller and Morrison respectively. Morrison had to limp out of the action and he was replaced by Felipe Teixeira.
The Portuguese player almost made an immediate impact with a curling shot which was only inches too high but Malouda should have made it 4-0 when he shot across the face of goal with only Carson to beat.
The Portuguese player almost made an immediate impact with a curling shot which was only inches too high but Malouda should have made it 4-0 when he shot across the face of goal with only Carson to beat.