The goal that proved to be the winner came in the 12th minute courtesy of Mexican striker Javier Hernandez.
Swansea's reply was positive and Scott Sinclair spurned a good chance to draw his side level. Nathan Dyer's shot was deflected to Scott Sinclair but with the goal at his mercy, the former Chelsea winger stood on the ball and Phil Jones intervened to clear.
The home side continued to search for an equaliser after the break, calling David De Gea into action on a couple of occasions. However, the Spanish keeper palmed away the efforts on his goal.
United went close to extending their lead in stoppage time, but full-back Jones saw his effort strike the post before rebounding to safety.
Nevertheless, the Red Devils held firm to ensure they left the Liberty Stadium with maximum points.