The Hammers earn just their third Premier League victory of 2015 over Sean Dyche's 10 men, who now look doomed to make an instant return to the Championship
Saturday's contest at Upton Park was settled when Michael Duff chopped down Cheikhou Kouyate in the 24th minute, with the defender unlucky to receive a red card for the challenge.
Noble sent goalkeeper Tom Heaton the wrong way from the resulting spot-kick and that was enough for West Ham to collect just a third league win of 2015.
Rock-bottom Burnley - who have three games remaining - are now eight points adrift of Leicester City and Hull City, while the gap to Newcastle United and Aston Villa is nine. If Burnley lose at Hull next weekend then the Lancashire will be back playing in the Championship next season.
A lack of goals has been Burnley's trouble and they have now drawn blanks in six straight matches, as Danny Ings and Ashley Barnes both saw good chances go begging in the first half.
Both teams had chances in a lively second half, but one goal was enough for Sam Allardyce's side to end a four-match winless run.
West Ham created the first opportunity in the third minute when Enner Valencia failed to put the finishing touch on Aaron Cresswell's teasing delivery across the six-yard box.
After a slow start Burnley came into the game and Ings put a free header from eight yards over the crossbar from Matt Taylor's excellent left-wing cross.
The flashpoint of the half came when Kouyate burst in to the penalty area and was clumsily tripped by Duff, resulting in referee Jon Moss producing a harsh dismissal for the veteran centre-back.
West Ham took full advantage as Noble put his penalty into the bottom right-hand corner.
With the extra man, West Ham turned the screw and Heaton made a stunning point-blank save from Valencia's close-range effort.
Burnley opted to stick with two strikers and Barnes - who returned from suspension in place of Lukas Jutkiewicz - saw a fantastic header from Kieran Trippier's corner saved expertly by the feet of Adrian.
The second half burst into life in the 58th minute when Morgan Amalfitano - brought into West Ham's starting XI at the expense of Matt Jarvis - curled wide from the left of the penalty area.
Burnley went up the other end and Taylor's low drive brought the best out of Adrian, while Nolan shot straight at Heaton in the area.
Stewart Downing then went into the book for simulation when he went to ground under Ben Mee's challenge in the penalty area, with Moss feeling any contact was not enough to warrant a foul.
The closing stages typified Burnley's season as they showed plenty of endeavour but little goal threat. And it was West Ham who created the better late chances - substitute Carlton Cole heading over before Kouyate had a late snatched shot cleared off the goalline.