The hosts came close to taking the lead after just two minutes when Phil Bardsley's cross almost crept in at the far post, whistling just wide.
© Getty Images
Sunderland had a great chance to equalise within three minutes of going behind when Ki Sung-Yeung's shot deflected into the path of Lee Cattermole, who could only send a weak shot straight at the keeper despite having time and space.
The visitors came close to doubling their lead in the 19th minute as Noble fed the ball to Matthew Taylor, who squeezed a shot narrowly past the post.
The Black Cats began the second half brightly, with Connor Wickham dragging a shot just wide moments after seeing his initial effort well blocked.
However, the Hammers were soon two up as a long free kick was chested down by Carroll for Mohamed Diame, whose shot from the edge of the box deflected past Mannone.
© Getty Images
It should have been game over shortly after the hour mark as Stewart Downing raced through on goal, but he saw his shot well saved with only the keeper to beat.
Sunderland took full advantage of that layoff, pulling a goal back just four minutes later as Craig Gardner fed a pass through to fellow substitute Adam Johnson, who curled his effort past Adrian and into the far corner.
They almost completed a quickfire comeback in the 68th minute as Ki's shot was fumbled by Adrian, who just about recovered in time to beat Wickham to the rebound.
The hosts threatened again two minutes later as Johnson picked out Fabio Borini inside the box, but the loanee striker could only slice his shot into the side-netting.
Sunderland continued to push for an equaliser in the closing stages and came close to getting it when a short corner was eventually rolled across the edge of the box for Marcos Alonso, who hammered a shot over the bar.
They had one last chance in stoppage time of the second half as Ignacio Scocco nodded over from Alonso's cross, but they couldn't conjure up a valuable equaliser, leaving Gus Poyet's men still four points from safety.