It was the first time that Burnley have lost three home Premier League games in a row since 2010, but they put up a good fight and made life difficult for the Red Devils in what was Sean Dyche's 100th top flight game in charge.
The Clarets opted for a more defensive five-man midfield with Scott Arfield replacing Sam Vokes and Ashley Barnes as a lone striker - who caused the Man United backline problems all afternoon.
Jose Mourinho praised Luke Shaw this week for how well he deputised for the suspended Ashley Young but it was Young who got the nod for his experience against resolute opponents after returning from his three match ban.
The game kicked off after a fitting tribute to Cyrille Regis who sadly passed away earlier this week.
James Tarkowski had a great chance to get an early goal for Burnley with a free header but he couldn't divert the ball goal-ward when he really should have done better.
It was then Manchester United's chance to snatch an early advantage. Young cut inside onto his favoured right foot and whipped an inch perfect cross in for Paul Pogba but the Frenchman's exquisite lob landed on the roof of the net.
There were few chances in the first-half but Young again provided a touch of brilliance down the left flank in the 42nd minute. He ran onto his own nutmeg on Johann Gudmundsson and then bent his shot just wide of the far post.
You might have thought the Red Devils would have been buoyed by the imminent arrival of Alexis Sanchez but their performance was rather flat and lacking the energy that Alexis will provide.
GOAL! Anthony Martial has broken the deadlock at Turf Moor!#BURMUN— 90min (@90min_Football) January 20, 2018
However, United got the breakthrough they were so badly craving in the 54th minute. Romelu Lukaku - who was non existent in the first 45 minutes - showed great awareness to find Anthony Martial on the left-hand side. The Frenchman then took his time to pick out his corner and hammered the ball home off the cross bar, leaving Nick Pope with no chance.
That was Martial's third goal in three appearances and although he is the most likely player to make way for the Chilean left winger in the starting line up, he was probably Man United's best player in this match.
Anthony Martial looking for his haters pic.twitter.com/E45HBfRhx7— FOX Soccer (@FOXSoccer) January 20, 2018
Burnley had their best chance of the game in the 70th minute when Gudmundsson raced down the right wing and fizzed in a dangerous ball along the corridor of uncertainty. Unfortunately for the Clarets, it took a nick off Chris Smalling's knee, diverting it agonisingly away from the lunging Tarkowski.
With the game coming to a close, Martial almost grabbed his second of the game as he used his pace to get past Phil Bardsley with ease and rifle a shot at Pope but the goalkeeper managed to palm the powerful effort away to safety.
The United win closes the gap to nine points behind Manchester City who face Newcastle United in the evening kickoff, while Burnley drop to eighth.