Substitute Juan Mata gave United the lead before Zlatan Ibrahimovic secured victory from what clearly appeared to be an offside position, after referee Mike Dean had controversially dismissed Sofiane Feghouli in the opening stages.
Despite its numerical advantage, United struggled to break down the home side for long spells and required a number of saves from goalkeeper David de Gea to keep West Ham out.
"I think it was not a very good performance," United manager Jose Mourinho said. "I think it's very difficult to play football in these circumstances, 48 hours," after both teams' previous fixtures.
West Ham manager Slaven Bilic said his club would appeal against the three-game ban Feghouli is facing after his challenge on Phil Jones.
"I was very surprised (at the red card), I wouldn't have been happy with a yellow. The more times I've seen it, it's the other way round, it was Jones who made a more dangerous tackle than Feghouli," Bilic said.
"We are gutted and frustrated, but I'm very proud of the team and I told them after the game that if we continue to work hard and play for each other, we don't have to worry, we're going to be very high on the table."
Manuel Lanzini brought two strong saves from De Gea before the break, while Michail Antonio was crucially denied by the Spanish goalkeeper when through one-on-one on the hour mark.
West Ham was made to pay just three minutes later with Mourinho's alterations proving crucial as Rashford beat two defenders before setting up Mata to finish from close range.
United wrapped up the victory 12 minutes from time when Ibrahimovic reacted first as the ball bounced off Ander Herrera to fire low and hard past Darren Randolph from just inside the box, with replays showing the striker had been marginally offside.
Antonio Valencia was set up at the back post after good work from Ibrahmovic and Henrikh Mkhitaryan, but was foiled by the sprawling Randolph, who prevented the ball from crossing his line by the narrowest of margins.
The rebound fell to Lingard with the goal at his mercy, but he could only direct his effort onto the post from two yards out, with West Ham somehow escaping as the ball bounced into Randolph's hands.
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