It was all Hammers in the first period, with good opportunities from Michail Antonio, Pedro Obiang and Noble before the skipper opened the scoring right at the death.
Burnley dominated in the second period, with good chances from Sam Vokes, Ben Mee and Scott Arfield but the Hammers managed to hold out for a crucial three points.
Slaven Bilic's side were in desperate need of the points, as they looked to pull themselves away from the relegation zone and they had a bright start to the clash.
The first came in the 15th minute, after patient build-up saw Noble cross the ball into a dangerous area. The ball looked to be heading to top-scorer Antonio, but the forward just couldn't quite get a connection on it.
He had another opportunity in the 33rd minute, when he was played off by Carroll; but his shot was straight at Tom Heaton.
Their best opportunity of the clash to that point came three minutes later. It was a powerful run from Obiang before letting rip, and it had Heaton scrambling before it went just wide of the mark.
Taking a crucial lead
The Hammers were turning the screw as they looked for a vital lead before the break. They couldn't have come closer than on the stroke of half-time, as Noble collected the ball from the short corner and he hit an excellent effort from the corner of the box, which cannoned back off the post.
Noble was given another opportunity in extra-time, when a defensive calamity saw Winston Reid bundled to the ground and Robert Madley pointto the spot. The tension inside the London Stadium was clear, and it looked like the opportunity had gone as Heaton saved the effort, but Noble lashed home the rebound.
One way traffic
Sean Dyche's men looked a different side coming out for the second period, and they had a host of decent opportunities to get the equaliser.
The 51st minute saw a half chance for Vokes, as the ball ricocheted into his path but he hooked it wide whilst hitting the ground.
The away side were at it again four minutes later when Mee headed the ball across goal, and Aaron Cresswell did well to clear with the ball from under his own crossbar.
Their best opportunity came just after the hour mark, it was a good run that carried Arfield into the area. His shot was blocked by Darren Randolph and floated perfectly in the air for Vokes, but his header went over with the goal gaping.
Right at the death
Both sides were battling right to the death, as the clocked clicked down at the London Stadium. Randolph was on a high after his save in Sunday's 2-2 draw with Liverpool, and he was it again in the 71st minute. It was a powerful free-kick from Arfield, but the Irishman was there to block.
The final opportunity came three minutes from time, with substitute Andre Ayew dancing through he defence and had Heaton beaten but it was well wide of the mark.