Coach Roger Cross fielded eight players with first team experience, but just couldn't find a way past an extremely young Blues side, who managed just two shots on goal all evening and rode their luck at the other end on the few occasions that Hammers threatened seriously.
Edouard Cisse was the chief culprit for the home side, wasting two glorious opportunities to break the deadlock. He burst clean through on goal in the 15th minute after dispossessing defender Watt but saw his shot pushed round the post by Chelsea keeper Rhys Evans.
The French midfielder then had the chance to make amends in the last minute of the game, but screwed his shot wide after Don Hutchison had released him in the penalty area.
Hammers's best chances in between Cisse's efforts had fallen to Irish defender Clive Delaney, who saw a powerful header cleared off the line in the first half, and Richard Garcia, whose fierce shot on the turn was superbly tipped on to the bar in the 79th minute.
A defence including the likes of Sebastien Schemmel, Christian Dailly and Gary Breen had few problems dealing with Chelsea's inexperienced and lightweight attackers, although midfielder Danny Jeffreys did threaten on the odd occasion.
The visitors - managed by former Upton Park player and assistant manager Mick McGiven - showed plenty of enthusiasm and desire, but Hammers will view this as two points dropped rather than a point gained.
Hammers: Bywater, Schemmel, Breen, Delaney, Dailly, Cisse, Noble
(Cohen 67), McMahon, Byrne, Hutchison, Garcia. Subs unused: Forde,
Chelsea: Evans, Kitamirike, Cousins, Watt, Huth, Nicolas, Oliveira (Smith 10), Woodards (Hollands 69), Kneissl, Boussoufa (Sentence 29). Subs unused: Pidgeley, Hudell.