The Red Devils were far from their best at Turf Moor, failing to create much besides their 54th-minute winner through a well-taken Martial finish.
United held on to their lead for the remainder to reduce the gap on leaders Manchester City, which now stands at nine points ahead of the Citizens' home match against Newcastle United this evening.
Less than a month on from their most recent meeting, which saw United recover from a couple of goals down to rescue a 2-2 draw at Old Trafford, the Red Devils again took time to get going.
Burnley edged things in the opening 45 minutes, in fact, getting in the half's only serious shot on target nine minutes in through Jeff Hendrick.
The midfielder pounced on a loose ball after Ashley Barnes had been denied by a Phil Jones block, only to pick out David de Gea - who has not conceded a single goal since the reverse meeting - down the middle.
Hendrick then chested down a cross into the box and scuffed his volley wide as the Clarets continued to have the better of things in an attacking sense, but the closest either side came to a breakthrough in a tight first half was via a curled attempt from Young.
After bursting into the box from the left, the full-back - back in the side following a three-match suspension - sent his shot drifting past the far post from a tight angle.
United had their key breakthrough goal less than 10 minutes into the second half, though, as Martial thumped the ball past Nick Pope via the crossbar after being picked out in the box by Romelu Lukaku's pass.
After conceding from their opponents' first on-target effort of the afternoon, Burnley did well to put United under some real pressure and almost had an instant response when winning a free kick 22 yards out.
Johan Berg Gudmundsson took on the set piece, getting it past the wall but seeing the ball hit the crossbar on its way through.
Burnley, who had not lost four games to start a calendar year since 1987 prior to today, again came close when Gudmundsson flashed the ball across goal just out of James Tarkowski's reach.
Barnes also glanced a free header wide from the Clarets' final real chance, while Martial was denied by a strong Pope hand at the other end late on as Jose Mourinho's men made it eight without defeat overall.
BURNLEY (4-2-3-1): Pope; Bardsley, Tarkowski, Mee, Taylor; Cork, Arfield (N'Koudou 81'); Defour, Gudmundsson, Hendrick (Vokes 81'); Barnes (Wells 89')
MANCHESTER UNITED (4-2-3-1): De Gea; Valencia, Smalling, Jones, Young; Matic, Pogba; Mata (Fellaini 72'), Lingard (Rashford 80'), Martial (Herrera 94'); Lukaku