The Swans, alongside Hull City, closed the gap in the fight for survival 24 hours earlier as they saw off the respective challenges of Stoke City and Watford.
But Sean Dyche's side were unab le to restore their cushion as United cruised to victory to extend their impressive unbeaten record.
The Clarets mastered their own 23-game unbeaten run last term to reach the dizzy heights of the Premier League as second tier champions.
But 20-time English champions Manchester United took things to the next level at Turf Moor by emulating that streak in the top flight.
Burnley boss Sean Dyche and his opposite number, Jose Mourinho, are essentially from differing schools of management but both have shown that their respective philosophies can work.
Dyche implanted a culture of familiarity and consistency, rarely changing his starting XI, as the Clarets embarked on an unbreakable spell on their climb to the top of the Championship.
Alternatively, the "Special One" showed that inconsistency in selection doesn't neccessarily mean that a club can't construct a positive pattern of form.
United's squad has been interchangeable, as has the system, with Zlatan Ibrahimovic, Marcus Rashford, Martial, Rooney, Jesse Lingard and Henrikh Mkhitaryan all vying for game time.
However, Mourinho has clearly found a formula that works - finding consistency within the inconsistency - as United are still pushing for a top five finish while competing in the latter stages of the Europa League.
The visitors didn't show any signs of exertion from their Thursday night quarter-final second leg against Anderlecht.
They were sharper, frighteningly quick on the break and punished the hosts when afforded any kind of space.
Former Red Devils youth player Robbie Brady threatened for the Clarets early on, swiping a volley wide of David de Gea's near post after former Old Trafford products Tom Heaton and Michael Keane has worked the ball in to his path from a set-piece.
United, though, kicked in to life and it was the pace of Frenchman Martial that seemed to be doing the damage.
The striker did well to pull the ball back to Ander Herrera after beating Keane to the bye line and in turn the midfielder picked out Wayne Rooney but defender Ben Mee took the sting out of the England international's shot.
The away side exposed the Burnley defence again minutes later, this time via an alternative route, when Marouane Fellaini leaped highest to meet Ashley Young's corner but Heaton was right behind the header.
The Clarets weren't without their chances, though they failed to test de Gea. The Spanish goalkeeper acknowledged the intervention of team mate Daley Blind on a number of occasions as the Dutchman often found himself in the right place at the right time.
The defender initially closed down George Boyd to block on the edge of the box after the winger had stepped inside on to his left boot.
And then his outstretched leg thwarted Andre Gray inside the six yard box after the striker looked to convert Jeff Hendrick's low cross. Mee then looped a header over the bar from the resulting corner.
United first signalled their intent to breach Burnley on the break in the 18th minute when Lingard zipped through the middle of the park and fed Rooney to his left hand side.
The Three Lions and United record-breaker's cross wasn't close to its intended target but momentarily threatened to drop in to the far corner when carried in the wind.
United opened the scoring just shy of the midway point in the first half and it was a product of Burnley's own undoing.
Brady conceded possession cheaply, Martial started the move in his own half, Joey Barton refused to use his experience to halt the counter and, after the former Monaco forward swapped passes with Herrera, he coolly slipped the ball past Heaton.
The lead was doubled in the 39th minute and, once again, the home side were guilty of inflicting the damage.
Brady handed the ball to Pogba, the £89m signing from Juventus slid the ball in to Martial and after Heaton repelled the striker's effort, Rooney reacted to roll the ball over the line.
The hosts had the opportunity to hit back just before the interval when Gray did impeccably well to turn and shoot inside the box but Eric Bailly blocked to divert the effort wide.
Chances were few and far between after the break with United's job done in the first 45 minutes of the fixture.
Pogba tried his luck from distance but Heaton fell to his left to palm the attempt to safety.
Keane, the subject of recent speculation linking him with a move away from Turf Moor in the summer, had the chance to put a dent in de Gea's Golden Glove hopes but the centre half guided Brady's free kick wide of the far post.
United almost extended their lead in the closing stages but the fingertips of Heaton denied Rashford after the substitute had raced goal side of James Tarkowski from Mkhitaryan's pass.
Rooney pulled an effort wide of the upright from 25 yards out before Pogba's thunderous drive kissed the outside of the post in the final minute.
One positive for the Clarets was the introduction of Dan Agyei for his senior debut on home soil and the striker went close to making his mark at this level when cushioning the ball on his chest, side-stepping Bailly before firing just wide of the far post.