The Hammers had a Felipe Anderson effort ruled out for offside before Man Utd took the lead on 19 minutes.
Pogba converted from the spot after Robert Snodgrass brought down Juan Mata inside the penalty area.
Manuel Pellegrini’s side equalised four minutes into the second half, when Manuel Lanzini’s cross finding Anderson, whose low finish beat David De Gea.
West Ham threatened to win it themselves with Michail Antonio hitting the bar before forcing a superb save from De Gea.
But United restored their lead with 10 minutes left, after Ryan Fredericks brought down Anthony Martial to hand Pogba the chance to score his second penalty, which he duly took.
United move up to fifth place on 64 points, two behind fourth-placed Chelsea, who travel to Liverpool on Sunday, with both sides having played 33 matches.
West Ham remain 11th with 42 points.
Man Utd: 21 Apr v Everton (A)
West Ham: 20 Apr v Leicester City (H)