It was a closed-off affair, to say the least, but the French forward profited from clever Romelu Lukaku hold-up play as part of a superb counter-attack to add to the Red Devils' perfect start to 2018.
Lingard and Pogba liven up the midfield
José Mourinho stuck with a 4-3-3 formation with Jesse Lingard the more advanced in front of Paul Pogba while Nemanja Matić was anchored in front of the defence, and Manchester United started the brightest.
Lingard and Pogba were bringing the ball out from midfield well as Ashley Young fulfilled his marauding full-back role to support Juan Mata and Anthony Martial in the final third, but a few weak crosses were all that came from early control.
Young whipped a dangerous ball over Burnley's defence from the left wing for Pogba to latch onto just after 10 minutes gone, but the Frenchman's innovative flick inside the box looped over the crossbar and onto the roof of the net.
Burnley had the first chance of the match though. Steven Defour's drilled free-kick from 50 yards out on the left met James Tarkowski unmarked at the back post, but the centre-back disappointingly headed a ranged effort way wide.
Young goes closest in a confined first half
The hosts' resolute defending continued throughout the first half as United's one-dimensional offensive play from the flanks wasn't proving successful and Romelu Lukaku was left starved of service for the most of the first half hour.
Ben Mee's 32nd-minute booking for a poorly-timed lunge on Jesse Lingard surprisingly didn't prompt more attacking intent from the visitors, but it was clear he'd be walking on ice from then on considering the pace he was up against.
Arguably the most promising United player on the pitch, Ashley Young, very nearly gave his side a lead in the latter stages of the first period.
He received the ball on the left flank and performed a couple of feints before nutmegging Jóhann Berg Guðmundsson as he entered the box and bending a fine effort just past Nick Pope's far post from a tight angle - the closest to an opener.
'Tony Martial scores again
The second half started just as the first finished, scrappy with limited quality on show. It was the away side doing the pushing though, and the increased tempo paid dividends less than 10 minutes after the restart.
Romelu Lukaku used his blistering pace to claim the ball on the right wing and did brilliantly well to hold it up under extensive pressure before finding Anthony Martial at the left side of the box.
And as he commonly does, the young forward kept his cool and smashed it high past a hopeless Nick Pope in front of the away faithful to extend his fine run of goal contribution.
Romelu Lukaku may have only scored once in 10 matches against top-half opposition ahead of this clash, but the part he played in that opener will deservedly earn plenty of plaudits.
Burnley respond well to going behind
Despite going behind, the home fans didn't tone down their support and Burnley very nearly replied just three minutes after the deadlock was broken.
Juan Mata clumsily took out Ashley Barnes right outside the Manchester United box and Guðmundsson curled his free-kick onto the crossbar and over - David De Gea was rooted to the spot.
The hosts thought they should've had a penalty on the hour mark after Guðmundsson's free header cannoned off Chris Smalling's midriff, but the referee and linesman weren't buying the claims one bit.
Burnley's best chance of the game came on 71 minutes after Lukaku lost the ball far too easily before Guðmundsson was in the thick of the action again and sprayed a tempting ball across the face of goal.
A heart-in-mouths kind of moment, but no teammates managed to get the vital touch and the search for a big equaliser was to continue.
Four United wins in four to start 2018
José Mourinho looked to solidify his team's lead and gain back control of the match by bringing on Marouane Fellaini for the uninspiring Juan Mata with 15 minutes to go.
Sean Dyche was next to flex his substitute options by bringing on Georges-Kévin Nkoudou for his debut after signing on loan from Tottenham Hotspur last week, as well as target man Sam Vokes.
There was no late drama, even though a hefty five minutes of stoppage time were added to the end match, and United made it four outings, four wins, with four clean sheets since the turn of the new year.
José Mourinho might be looking to give imminent January signing Alexis Sánchez his debut when Manchester United travel to Yeovil Town in the FA Cup fourth round on Friday night.