The result left United second on 57 points from 29 games, 14 behind Manchester City who have played a game more, while West Ham stayed fifth on 48 points from 28 matches, three behind fourth-placed Chelsea with a game in hand.
United were twice denied by the woodwork and by a pair of fine saves from visiting goalkeeper Lukasz Fabianski but they ground out their victory as Dawson turned an inswinging Bruno Fernandes corner into his own net in the 53rd minute.
Fabianski tipped a Mason Greenwood shot onto the outside of the post in the first half and, after the hosts took the lead, clawed a first-time shot by Fernandes out of the bottom right corner.
Greenwood rattled the woodwork again after a darting solo run from his own half in the 76th minute as the home side looked to exploit gaps with West Ham throwing men forward.
Next up, United will play the second leg of their Europa League round-of-16 tie with AC Milan on Thursday before visiting Leicester in the FA Cup quarterfinals on Sunday.