Martial’s second-half strike was one of only two efforts on target from the visitors in a display that was effective rather than entertaining.
After a dour first half, Martial fired in off the underside of the bar after good work in the build-up from Romelu Lukaku.
Burnley pressed forward in search of a leveller, coming closest when Johann Berg Gudmundsson’s free-kick hit the bar.
But United goalkeeper David de Gea rarely looked like blotting his side’s fifth successive clean sheet despite a siege on his goal in the final five minutes.
Manchester City defeated Newcastle 3-1 later on Saturday to maintain their advantage over their city rivals.
As United extended their unbeaten run in the league to eight games, it was a scoreline – rather than performance – for their travelling fans to enjoy.
In front of club legends Sir Alex Ferguson and Sir Bobby Charlton, United took the points in a style in keeping with manager Jose Mourinho’s regular modus operandi, rather than the club’s dynamic, attacking traditions.
The apparently imminent arrival of industrious Arsenal forward Alexis Sanchez will surely jump-start an attack that lacked pace, invention or a single shot on target in the first half.
But, on this occasion, the individual quality of United’s squad emerged for just long enough to deliver the decisive goal.
The previously pedestrian Lukaku came to life, barging aside Ben Mee and Steven Defour to secure the loose ball before sweeping it across the field to find Martial, who expertly stood up Phil Bardsley before scoring.
