The French forward powered an effort past Nick Pope nine minutes after the restart for the match-winning moment - one of just two on-target attempts for the visitors.
Despite struggling to truly get going, the Red Devils were not put under a great deal of pressure at the other end and held on to extend their unbeaten run to eight matches.
Relive how the 90 minutes of action unfolded in Lancashire with Sports Mole's live text coverage below.
STARTING XI: De Gea; Valencia, Smalling, Jones, Young; Matic, Pogba; Mata, Lingard, Martial; Lukaku
SUBS: Romero, Rojo, Shaw, Fellaini, Herrera, McTominay, Rashford
STARTING XI: Pope; Bardsley, Tarkowski, Mee, Taylor; Cork, Arfield; Defour, Gudmundsson, Hendrick; Barnes
SUBS: Lindegaard, Lowton, Nkoudou, Vokes, Westwood, Wells, Long
Sean Dyche: "If you look, every single club, other than the top six, have had various spells like we've just had. Some, who are struggling a little bit down at the bottom, have had three or four spells. That's how difficult the Premier League is. When we had a really good run I said the margins were tight and we were on the right side of them. We want to get back to doing that."
Jose Mourinho: "I think clearly everybody knows that we are there [with Alexis Sanchez] and especially when the Arsenal manager speaks so clearly like he did, there is no point in trying to hide or deny it. But [the transfer] is not done at all, so in this moment [Henrikh] Mkhitaryan is our player, Sanchez is an Arsenal player and with a match tomorrow, I want to switch off and focus on the most important thing, which is the match."
BURNLEY SUBS: Lindegaard, Lowton, Nkoudou, Vokes, Westwood, Wells, Long
MANCHESTER UNITED SUBS: Romero, Rojo, Shaw, Fellaini, Herrera, McTominay, Rashford