The 20-year-old muscled himself into space to head Robert Snodgrass's corner powerfully past the keeper for his second Premier League goal.
The home side survived the disruption of an injury to Aaron Cresswell, who was replaced by Arthur Masuaku, to extend their lead from the spot after Florian Lejeune brought down Javier Hernandez just before the break.
The ever-reliable Noble rifled his shot high into the corner for his 19th successful penalty for the club out of 21.
Although Newcastle improved after the break, when Paul Dummett came on in a rejigged formation, they seldom threatened beyond a Salomon Rondon shot that hit the post late on. The win moved West Ham to ninth in the table.
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(Reporting by Neil Robinson; editing by Clare Fallon)
