Ole Gunnar Solskjaer's side kept its first clean sheet in the league since September and successfully broke down Burnley's packed defence with some patient attacking play, forcing the opener just before half-time and doubling the lead in second-half stoppage-time through Marcus Rashford.
After losing 2-0 at Watford in its previous away game, United delivered the improvements it sorely needed at the back while dominating possession in midfield despite the absence of Paul Pogba, who was rested after his return to fitness, as it built on a 4-1 win over Newcastle United on Boxing Day.
Burnley was looking to make it three league wins from four but failed to recover from an error by full-back Charlie Taylor that led to Martial's strike, squandering a flurry of late opportunities before Rashford scored his 12th league goal of the season.
Rashford stung Nick Pope's palms with a free-kick from 30 yards in a cagey opening spell in which the England forward also hit the outside of the post with a low, first-time shot from the edge of the box.
The ball fell to Martial little more than six yards out after 33 minutes, but the France international's effort was crucially blocked on the line by former United defender Phil Bardsley.
Having kept the visitors at bay for 44 minutes, Burnley self-destructed when Taylor fumbled possession and allowed Andreas Pereira to feed Martial, who took a touch and stroked a low shot past Pope into the bottom-right corner.
Martial had the ball in the net again four minutes into the second half, but the goal was disallowed for a foul on James Tarkowski and Burnley almost equalised when Bardsley's angled drive drew a diving save from David de Gea.
Harry Maguire brought down Jack Cork on the edge of the United box to present substitute Johann Gudmundsson with a chance from a free-kick, but the Iceland international blazed over the crossbar.
Burnley threatened an equaliser throughout a frenetic final 15 minutes only to be caught out right at the end when Daniel James sprinted over the halfway line in possession and squared to Rashford, who rounded Pope before scoring his second in as many games.