The Hammers will be without star forward Carroll as he serves a three-match ban for violent conduct, and is set to miss the upcoming games against Aston Villa, Norwich and Southampton.
But Noble is still confident that West Ham will be able to build on last Saturday's 2-0 victory over Swansea City.
The 26-year-old told the London Evening Standard: "Andy being suspended is obviously a massive miss for us.
"For him to be back the way he was and then to miss him for another three games is going to be tough but we have to dig in and get results without him.
"Everyone will have to play their part to try and get us three points away from home.
"We're three points off eleventh, so getting ourselves back in that pack with the win against Swansea was important.
"It would be good to go to Aston Villa and get the three points before another winnable home game against Norwich City.
"We're looking up rather than down at the moment and even though we're in the bottom three, I'm sure we can claw our way back."