Noble’s first-half spot-kick, which saw Michael Duff sent off for a foul on Cheikhou Kouyate, left the Clarets bottom of the table – eight points from safety with just three games to go.
And wins on Saturday for relegation rivals Aston Villa, Sunderland and Leicester means that Sean Dyche’s side will now almost certainly be returning to the Championship next season.
It was a slow start at Upton Park, with no real chances until the 20-minute mark when Danny Ings, now without a goal in 10 matches, should have done better than head over the bar after losing his marker in the box.
But just a couple of minutes later, West Ham were ahead through Noble’s penalty, Kouyate brought down by Duff’s hanging leg following a jinking run into the box.
Duff was dismissed by referee Jon Moss, much to the fury of his team-mates, and Noble sent Tom Heaton the wrong way with the resulting spot-kick.
It could have been worse were it not for Heaton, who first saved superbly from a close-range effort by Enner Valencia, before making a smart stop at his near post after a powerful shot from Morgan Amalfitano to keep the deficit to just a single goal at the break.
West Ham’s intensity dropped, although they did have their chances through Amalfitano – whose curling effort flashed just wide – then captain Kevin Nolan, who hit the keeper from 12 yards when he was free in the area.
Former Hammers midfielder Matt Taylor drew a smart stop from Adrian, while Ings was denied by the side netting.
But though the game opened up, Burnley could not rally in the dying moments and could now be relegated next weekend away at Hull.
West Ham: Adrian 7, Jenkinson 6, Collins 6, Burke 6, Cresswell 7, Amalfitano 7, Noble 7, Kouyate 7, Nolan 6, Downing 6, Valencia 6
Subs: Nene 5, Cole 5
Burnley: Heaton 7, Trippier 6, Shackell 6, Duff 4, Mee 5, Boyd 5, Arfield 5, Jones 5, Taylor 5, Barnes 5, Ings 5
Subs: Keane 5, Sordell 5, Wallace 5
Man of the match: Mark Noble