With leaders Manchester City facing Newcastle United later on Saturday, United have closed the gap to nine points, for a few hours at least.
Martial blasted home off the under-side of the bar in the 54th minute after brilliant work from Romelu Lukaku, who held off two challenges before finding the Frenchman with a sweeping cross-field pass.
Burnley, who slip to eighth with the loss, fought hard for an equaliser and Icelandic winger Johann Berg Gudmundsson struck the bar with a free kick.
"It was certainly a battle and that is what you get when you come to Turf Moor - it is about character and we showed that in abundance today," said United defender Phil Jones.
Manager Jose Mourinho said after the game that he "thinks" Arsenal forward Alexis Sanchez will complete his move to Old Trafford. The disappointment of Arsenal fans will have been eased by their attacking display against Crystal Palace in a 4-1 win at the Emirates.
Arsenal, who hadn't won in their last five games in all competitions, struck their four goals inside the first 22 minutes.
Nacho Monreal headed in from a Granit Xhaka corner and then turned provider for close range goals by Alex Iwobi in the 10th and central defender Laurent Koscielny three minutes later.
French striker Alexandre Lacazette made it four for the home side in the 22nd minute, collecting a clever backheel from Mesut Ozil in the area before turning to curl his strike around outstretched Palace keeper Wayne Hennessey.
Palace captain Luka Milivojevic pulled one back for the visitors in the 78th minute.
Eden Hazard scored twice as Chelsea enjoyed a 4-0 victory at Brighton and Hove Albion -- giving Antonio Conte's side their first league win of 2018.
Hazard and Willian struck inside six minutes to set them on their way. With Brighton pushing forward, Hazard brilliantly netted a third 13 minutes from time before Victor Moses made it four late on.
The win took Chelsea above Liverpool -- who play on Monday -- into third. With one win in 13 league games, Brighton are 16th, just two points above the relegation zone.
Stoke City got themselves out of the bottom three, with new manager Paul Lambert enjoying a 2-0 win over Huddersfield Town.
Fellow strugglers West Bromwich Albion picked up a useful point at Everton in a 1-1 draw.
