Premier League - Drogba leads Chelsea to win
However, Fernando Torres failed to end his goal drought despite Chelsea spending much of the last 10 minutes trying vainly to set the striker up. Torres did put the ball in the net but saw it rightly chalked off for offside.
The defeat was the first for West Brom in seven matches under new manager Roy Hodgson, who had previously plotted the downfall of Carlo Ancelotti's side this season when his Liverpool team beat them 2-0 at Anfield back in November.
That match was won courtesy of two goals from Torres, but the Spaniard was named as a substitute for the third Premier League fixture in a row.
Just as in Tuesday's Champions League defeat to Manchester United when he had to look on from the bench as his half-time replacement Drogba outperformed him, the British record signing once again saw the Ivorian striker shine in a side so accustomed to playing to his strengths.
However, it was not until Chelsea were a goal down that they truly clicked into gear. They were sparked into action after Chris Brunt and Jerome Thomas combined to release Odemwingie through the middle. The Nigerian had lost the ball-watching John Terry and surged forward to lift a deft finish over Petr Cech.
After that, Chelsea began to make their superior possession count, and they were level within five minutes when Florent Malouda's low cross just evaded the fingertips of Scott Carson. Full-back Nicky Shorey got his control all wrong inside the box and the ball rebounded invitingly for Drogba who lashed home his 13th of the season, and for his eighth goal in seven games against the Baggies..
The turnaround was soon complete when Drogba's low shot from range was spilled by Carson and Salomon Kalou - starting ahead of Nicolas Anelka - dispatched a clinical low finish into the far corner.
David Luiz - cup-tied for Tuesday's European defeat - had an effort blocked at the near post and Carson tipped a well-struck Lampard free-kick over the bar, but the England grabbed his first Chelsea goal in seven games just before the break.
Drogba had dropped deep to dispossess Odemwingie and he sent Malouda away down the left flank. The Frenchman pulled the ball back to Lampard, who had time enough to shift and side-foot past Carson for his seventh league goal of the season.
Chelsea's dominance continued after the break. Drogba had a speculative effort from long range land on the roof of the net, Malouda had a low drive palmed away by Carson and Kalou had a shot from inside the box deflected on to the crossbar.
West Brom had the occasional sight of goal, with Chris Brunt twice firing narrowly wide of the top corner and Odemwingie scooping an effort over the bar, but Chelsea were always favourites to score the next goal.
With that in mind, Ancelotti brought Torres on with eight minutes to go in the hope of breaking his Chelsea duck at the 12th attempt. Soon enough the Spaniard latched on to a through ball from Malouda, rounded Carson and coolly knocked the ball into the empty net, only to see the linesman's flag raised for offside against him.
The Spaniard then miscued a first-time shot from Malouda's free-kick as his team-mates desperately tried to set him up for another chance, but yet another blank for the £50 million man was the only thing which blighted an otherwise convincing response to their Champions League disappointment.