The Hammers took an early lead in the ninth minute after England international Carroll nodded past Sunderland goalkeeper Vito Mannone at the Stadium of Light.
It puts more space between us and the bottom three and we can look up towards Stoke in tenth place now
The 25-year-old also played a part in West Ham’s second with the lay-off for Mohamed Diame before Sunderland substitute Adam Johnson scored to ensure a nervy final 25 minutes.
“Yes I think we are [safe],” Carroll told Sky Sports.
“Obviously we have to take every game as it comes. Obviously the teams are winning down below us. But all the lads are positive about staying in the league and I think we are safe.”
Hammers captain Kevin Nolan challenged the east London side to catch 10th-placed Stoke City after their win.
“We knew it was going to be tough. We dealt with them well,” Nolan added.
“Adam Johnson made a great impact when he came on – a fantastic goal.
“But I felt we were quite comfortable throughout the game. You get no easy games especially when they are scrapping for their lives.
“We knew we would go 12 points clear of them if we won and it was massive.
“It was bigger for them more than us tonight. We had a game plan and it worked.
“It puts more space between us and the bottom three and we can look up towards Stoke in tenth place now.”
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