Man United remained in fourth place with 63 points from 34 matches, one point ahead of Liverpool, who played 35 games.
West Ham raised its tally to 37 points in 15th place, seven points ahead of the relegation zone.
The visitors had their best chances in the first half, but they paid the price for De Gea’s mistake.
Saeed Bin Rahma kicked off from the middle of the field and hit a weak ball, which the Spanish goalkeeper failed to stop, and it settled into the net.
West Ham demanded a penalty kick after a handball against Manchester United defender Victor Lindelof, but the video assistant referee confirmed the validity of the referee’s decision to continue playing.
De Gea saved a chance from Tomas Sochik in the second half, before the West Ham player hit the net with a header, but the referee ruled out the goal for offside in the 73rd minute.
West Ham goalkeeper Lukasz Fabianski saved two chances from Marcus Rashford and substitute Anthony Martial after the break.
Despite counting 10 minutes of stoppage time, United failed to make amends, suffering a second successive league loss.
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