Carroll hit in his sixth goal in six games and Noble tucked away two penalties to move the Hammers to within four points of fourth-placed Arsenal.
The duo are likely to be overlooked for England's Euro 2016 squad this summer, but they could still be embarking on a European adventure with West Ham next season instead
They could hardly have asked for more obliging opposition, though, with Watford safe from relegation and gearing up for Sunday's FA Cup semi-final
The Hornets clearly had one eye on their Wembley meeting against Crystal Palace with boss Quique Sanchez Flores making seven changes
Yet they still pulled one back through Sebastian Prodl and had a late penalty of their own saved before striker Nordin Amrabat was sent off in injury time
In a one-sided first half West Ham took the lead in the 11th minute when Dimitri Payet cut inside from the byline and curled the ball perfectly for Carroll to steer past Heurelho Gomes with his left shin
They should have been home and hosed long before Noble struck the second, but Carroll headed narrowly over, Payet twice shot over the crossbar and Gomes saved well from Diafra Sakho
West Ham had been growing increasingly annoyed at having been awarded just one penalty all season before Sunday's stormy draw at Leicester, but they have now been given three in two matches
Referee Mike Dean first pointed to the spot in the 45th minute after Jose Holebas climbed all over Cheikhou Kouyate as they contested a high ball in the area
Gomes memorably saved two penalties from West Brom's Saido Berahino on Saturday, but was unable to record a hat-trick as Noble sent him the wrong way from the spot
Manuel Lanzini then missed a sitter to make it three in first-half stoppage time when he sliced Sakho's cross wide from eight yards
Watford could have hauled themselves back into the match eight minutes after the interval, but instead a nightmare 60 seconds for midfielder Almen Abdi put West Ham out of reach.
In a rare Hornets attack Abdi slid in to meet Jose Manuel Jurado's cross six yards out but somehow sent it over the crossbar.
Moments later the Switzerland international was back in his own area, where his clumsy foul on Michail Antonio gifted Noble his second and West Ham's third .
Prodl volleyed home a consolation from a Steven Berghuis cross in the 64th minute, and in stoppage time Dean awarded a third penalty of the night when Angelo Ogbonna grappled with Prodl
However, Adrian saved substitute Troy Deeney's spot-kick and moments later Amrabat saw his hopes of a Wembley appearance ended by a second yellow card for a foul on Winston Reid
TWEET OF THE MATCH
"PENALTY KING: @Noble16Mark has now taken 28 penlaties for West Ham - and scored 25! [3-0]" - West Ham's official account pays Mark Noble his spot-kick dues.
RATINGS
West Ham
Adrian: 7
Michail Antonio: 7
Winston Reid: 6
Angelo Ogbonna: 6
Aaron Cresswell: 7
Mark Noble: 7
Cheikhou Kouyate: 7
Manuel Lanzini: 7
Dimitri Payet: 8
Diafra Sakho: 6
Andy Carroll: 7
Subs
Victor Moses: 6
Emanuel Emenike: 6
James Collins: 6
Watford
Heurelho Gomes: 6
Juan Carlos Paredes: 5
Sebastian Prodl: 6
Craig Cathcart: 6
Jose Holebas: 5
Mario Suarez: 5
Valon Behrami: 5
Adlene Guedioura: 5
Jose Manuel Jurado: 6
Almen Abdi: 4
Nordin Amrabat: 4
Subs
Steven Berghuis: 6
Ikechi Anya: 6
Troy Deeney: 6
STAR PLAYER
Dimitri Payet
By no means the Frenchman's most spectacular outing of the season but virtually every West Ham attack went through Payet, whose close control and eye for a pass are pivotal to their style of play.
MOMENT OF THE MATCH
To call it a 'shinner' would be doing Andy Carroll a disservice as the striker was in mid-air as he guided West Ham's opening goal past Heurelho Gomes for his sixth in six matches.
VIEW FROM THE BENCH
As predicted, Watford boss Quique Sanchez Flores made changes, seven in total, with one eye on Sunday's FA Cup semi-final
Skipper Troy Deeney started on the bench and fellow striker Odion Ighalo did not even travel
Hammers chief Slaven Bilic welcomed back Diafra Sakho in attack after a knee injury.
MOAN OF THE MATCH
It was hard not to feel sorry for Hornets midfielder Almen Abdi, whose miss at one end and - 60 seconds later - foul at the other effectively cost his side any chance of getting something from the game.
WHO'S UP NEXT?
West Brom v West Ham (Barclays Premier League, Saturday, April 30)
Crystal Palace v Watford (FA Cup semi-final, Sunday, April 24)
Source : PA
Source: PA