Andy Carroll, who has hit a purple patch at the perfect time as he bids to win a spot in Roy Hodgson's England squad for Euro 2016, hit his sixth goal in last five league games with an awkward looking volley to open the scoring after 11 minutes.
Noble scored twice from the penalty spot either side of halftime before Sebastian Proedl swept home after 64 minutes to reduce the deficit. Any hope of a late fightback, however, was effectively snuffed out when Troy Deeney's late penalty for the visitors was saved.
Watford's Nordin Amrabat was sent off for a second booking at the death as West Ham extended their unbeaten league run to nine games to stay sixth in the table and on course for a Europa League qualifying spot.
(Reporting by Toby Davis; editing by Brian Homewood)
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