Carroll opened the scoring after just nine minutes before setting up what proved to be the winner five minutes into the second half to give the Hammers a victory that lifts them 11 points clear of the relegation zone.
The 25-year-old insisted that he would still take things on a game-to-game basis but does not believe that his side will be dragged back into the relegation mire with just six matches left.
"It's a great feeling to win against a team like Sunderland but all that matters is the three points," Carroll told Sky Sports News.
"Personally I need to keep fit, keep involved and keep scoring goals. We need to take things as they come, the teams below us are picking up points but we're confident of staying in the Premier League and I think we're safe now."
While the Hammers are now 12 points from the bottom three, Sunderland remain in trouble, sitting in 19th place and four points adrift of safety.