Slowly, the Baggies ran out of stream and Diafra Sakho was inches away from connecting with Aaron Cresswell’s cross as the Hammers began to assert their authority after the hosts’ brash start.
Dimitri Payet became more influential and it was the France international who eventually unlocked Albion to help the Hammers open the scoring after 34 minutes.
The hosts struggled to clear from a West Ham corner and when a fortunate ricochet broke for Payet he found Kouyate to head in at the far post.
It underlined the Hammers’ extra cutting edge and they doubled their lead in first-half injury time as the hosts were caught cold again.
Kouyate was sent scampering down the left and raced unchallenged into the area to tee up Noble, whose survived a poor first touch to stab in the corner.
It was another sloppy goal for the Baggies to concede as they were left to rue Gardner’s early misses with the Hammers increasingly comfortable.
Slaven Bilic’s side had more quality, the gulf highlighted by Rondon’s weak header being easily gathered by Adrian, before Saido Berahino missed a glorious chance after 64 minutes.
The striker had only been on the pitch for less than 90 seconds when Rondon cushioned a cross into his path but he could only drag an effort inches wide from 15 yards.
Fellow substitute Sandro shot wide but the game drifted, playing into West Ham’s hands as they batted any limp attacks from the hosts.
And they wrapped up victory with 11 minutes left when Noble bagged his second.
Payet was allowed the chance to run at the Baggies defence and tee up Carroll to cross for Noble to adjust his body and smash a 10-yard volley into the roof of the net.