Allardyce's team dominated much of the match and created numerous chances in the first half without finding the net, before Mohamed Diame gave them the lead two minutes into the second period.
Fulham were slightly improved after the break, but West Ham wrapped up the win with late goals from substitutes Carlton Cole and Joe Cole.
The two sides went into the match on 10 points and very much embroiled in the battle at the bottom of the Premier League table, and Allardyce believes that victory will prove to be a big boost.
"It was a big win for us," he said. "The pressure is always on when you don't get results and slide towards the bottom three.
"When you come up against your nearest rivals in the table, with both sides on 10 points, it is a six-pointer so putting three points between us and Fulham is obviously a very healthy situation for us.
"I always believed if we created chances we would score goals and we did that today."
Fulham manager Martin Jol has come under pressure after his side's slump, and the defeat is only likely to see that increase, but Allardyce does have sympathy for the Dutchman.
"You can't not feel for each other when things aren’t going your way," he added. "It's a lonely job when you’re trying to find a solution to poor results.
"It's one where people don't really understand what you’re going through because they have not experienced it.
"The people you interact with on a regular basis have to help you because you're in a position where you take all the heat. That's the job. If it’s going bad it is your responsibility."
‘Maybe I’ll be signed by Wrexham, who knows? English football would suit me – if an offer came, I wouldn’t think twice, I’d go immediately’ Euro 2024 cult hero reveals his UK dream
‘Managing Leeds? It was an option that appeared, but it wasn’t the right timing. I decided it wasn’t a good idea to leave the club I was at mid-season’: Premier League boss admits to turning down opportunity to replace Jesse Marsch in 2023
