West Ham started the better of the two sides and attacked from the very start.
Fulham tried to play their way back into the game after the early setback and debutant Kieran Richardson came close on seven minutes.
The away side continued to press, though West Ham were unlucky not to score on nine minutes when Mohamed Diame's shot rattled the crossbar.
Carroll had an effort on goal blocked by the Fulham defence after 20 minutes as West Ham looked to double their advantage.
Sam Allardyce's side did manage to grab a second goal on 29 minutes when Winston Reid headed in Matthew Taylor's corner.
Taylor then made it 3-0 on 41 minutes to round off a dominant first half for the home side.
The half-time whistle blew with West Ham deservedly in front.