Mateja Kezman finally broke his Chelsea goal duck with the winner against a spirited West Ham in the Carling Cup.
Mateja Kezman (centre) celebrates his first goal for Chelsea
The striker was making his 13th outing for Chelsea and he arrowed in a second half shot past keeper James Walker.
The Serbian, who had to be treated for a head injury caused by an object thrown by a fan, missed a succession of good chances before his goal.
Walker saved a Frank Lampard penalty late on while an Anton Ferdinand header hit the bar for the Hammers.
Chelsea had looked ominously dangerous in the pouring rain at Stamford Bridge and Cole was particularly bright against his former club.
But he and his Stamford Bridge team-mates were being continually thwarted by Walker.
He was quickly called into action when he pulled off an excellent save to keep out a Kezman shot which was deflected off Hayden Mullins.
Walker did well to gather a Cole shot soon after and then watched as Geremi blazed a shot over his bar after Arjen Robben had found the midfielder with a cut back.
West Ham had survived Chelsea's opening assault and were starting to grow in confidence as the game progressed.
Adam Nowland struck a true shot for the visitors but was just over before a mix-up in the Hammers defence allowed Cole a shot on goal which Walker again saved.
At the other end, Nigel Reo-Coker worked himself into a promising position only to see a strike blocked by William Gallas when he might have squared a pass to better placed to team-mates.
Kezman's woes in front of goal showed no signs of abating, especially when he headed wide from close range.
Marlon Harewood clattered into Blues keeper Carlo Cudicini just before half-time as the Championship side more than held their own against their illustrious rivals.
A Kezman effort was cleared off the line by Rufus Brevett at the start of the second half as his frustrations increased.
But, after 611 minutes without a goal, Kezman broke his duck for Chelsea when his shot beat the despairing dive of Walker.
Robben won a penalty late on which another former Hammer in Lampard took only to see Walker again pull of an excellent save.
West Ham had put up a spirited display and were almost rewarded when Ferdinand struck the bar with a header.
But Chelsea held on to progress to the fourth round.
Chelsea: Cudicini, Paulo Ferreira, Ricardo Carvalho, Gallas,
Babayaro, Geremi, Parker (Lampard 68), Tiago, Kezman,
Robben (Gudjohnsen 82), Cole (Duff 64).
Subs Not Used: Cech, Huth.
Goals: Kezman 57.
West Ham: Walker, Mullins, Ferdinand, Repka, Brevett,
Etherington (Rebrov 85), Nowland (Noble 65), Reo-Coker, Lomas,
Harewood, Zamora (Hutchison 83).
Subs Not Used: Bywater, Cohen.
Ref: A D'Urso (Essex).