On loan from Watford, Dawson has been outstanding in the Hammers’ push for a top-four slot but settled the contest in the wrong way at the start of the second half as he turned Bruno Fernandes’ corner into his own net under pressure from Scott McTominay at the near post.
Marcus Rashford and Mason Greenwood also went close in a one-sided contest as West Ham failed to threaten until Said Benrahma’s introduction after an hour.
When the visitors finally pushed forward, Luke Shaw got in the way of Tomas Soucek’s goalbound shot as United kept a fourth successive clean sheet for the first time under Ole Gunnar Solskjaer.
The result kept United in second but with the teams from fourth to seventh all failing to win this weekend, they now have a nine-point advantage over West Ham, who remain fifth, but do have a match in hand.
Solskjaer will have been relieved to get the victory after conceding injury-time equalisers against Everton and AC Milan at Old Trafford in recent weeks.
It was not the greatest performance and United’s first-half frustrations were summed up when skipper Harry Maguire bellowed an instruction to his team-mates that they needed to “liven up” and “stop giving the ball away”.
With David Moyes setting up his side with wing-backs and three holding midfielders, West Ham provided a puzzle United struggled to solve.
Aside from Rashford glancing a header wide and Greenwood having an effort tipped on to a post and out for a corner by Lukasz Fabianski, United’s passing lacked both the accuracy and crispness to move the visitors out of their containment structure.
In addition, as Rashford kept pulling wide and Fernandes tended to drop deep, neither McTominay nor Fred made runs into the box that might have allowed them to threaten the visitors’ goal.
United were better after the break and Fabianski produced another excellent low save to deny Fernandes before Greenwood struck a post with a powerful shot at the end of a burst into the box.
The result means Moyes remains without a win as an away manager at Old Trafford, a sequence that now extends to 11 defeats in 15 attempts in all competitions.
It is now 17 games since West Ham last beat Manchester United away from home, going right the way back to the famous Carlos Tevez winner that kept the Hammers up in 2007.
The win marks the first time United have done double over the Londoners since 2013-14, when Moyes took over from Sir Alex Ferguson.
West Ham never really looked capable of winning the game. The absence of Pablo Fornals through injury cost them an element of creativity they did not introduce until Benrahma came on, while Maguire and Victor Lindelof were assured on the rare occasions West Ham had a chance to create any aerial problems.