Chris Smalling scored 22 seconds after his introduction to put United ahead and the substitute put United 2-1 up after Danny Ings' spectacular diving header.
Burnley, whose team cost a fraction of Louis van Gaal's star-laden XI, had United on the ropes at times in the first half and comedy defending from Smalling and Ander Herrera should have resulted in an equaliser, but David de Gea saved the Red Devils once again.
Finally, Robin van Persie put the game to bed with an 82nd-minute penalty after Angel Di Maria was fouled in the box.
All the focus in the build-up to Wednesday's match had centred on Van Gaal's bizarre press conference stunt, which was designed to shoot down Sam Allardyce's claims about United's style of play.
Allardyce had referred to Van Gaal's team as "long-ball United" and Van Gaal responded by producing a dossier of statistics which supposedly undermined the Hammers manager's claims at a press conference on Tuesday.
There were few long-balls on Wednesday night - Marouane Fellaini did not step off the bench to add an aerial threat - but for large parts United were still not at their free-flowing best.
The fact that Wayne Rooney was moved to the base of United's midfield diamond following an injury to Daley Blind did not help.
Still, Van Gaal will point to the Barclays Premier League table, which states his team are now third - one point ahead of Southampton and two in front of Arsenal.
United were criticised for a lack of creativity on Sunday, but Van Gaal stubbornly stuck with the same front six that looked completely blunt at West Ham.
The only changes for United came at the back where Paddy McNair and Jonny Evans replaced Antonio Valencia and Luke Shaw.
United fans applauded Michael Keane as his name was read out from the Burnley team-sheet, but the centre-back almost scored against his old club after just two minutes.
Scott Arfield's corner cleared a number of United bodies in the box, but Keane's effort flew wide.
United heeded the warning
The man who jumped ahead of Keane in the pecking order at United, McNair, almost celebrated his new contract with a maiden goal but Tom Heaton awkwardly palmed his shot wide.
Just before Rooney took the corner Smalling came on for Phil Jones and 22 seconds later he headed United ahead.
From Rooney's corner, Radamel Falcao knocked Van Persie's flick-on back across the box and Smalling headed home.
The United fans hoping their team would go on to establish a comprehensive lead were to be disappointed
The hosts' lead lasted just six minutes.
Di Maria gave the ball away and Kieran Trippier took advantage, swinging in a cross to the back post where Ings headed in superbly after drifting away from Smalling.
Burnley upped the pressure and would have taken the lead had De Gea not saved Ings' low shot with his feet
Ashley Barnes cracked a low curling shot just wide
Evans hoofed the ball up in the air inside his own box moments later
United were rattled.
Whatever they tried in attack - whether it be long passes, short passes or medium-sized ones - it did not work
Burnley, promoted to the top-flight on a shoestring, were upstaging Van Gaal's expensively-assembled team.
Michael Kightly even nutmegged Rooney.
Van Gaal then took the bizarre decision to drop Rooney, the club's third-highest goal scorer of all-time, back to the base of the diamond when Blind came off through injury and was replaced by Herrera.
Smalling saved United's blushes just before the break when he scored his second.
Again the goal came from a corner
Rooney and Di Maria exchanged passes before the Argentinian whipped a pin-point ball to the back post where Smalling rose above his marker and nodded in.
United still looked shaky at the back after the break
Evans looked uncomfortable on the ball.
Smalling looked at Herrera, expecting the Spaniard to clear George Boyd's soft cross, but he did not and the ball fell straight to the feet of Ings
Not for the first time this season, De Gea came to the rescue, deflecting the ball over the bar.
Rooney's evening got worse when he was booked for barging into Ings
Falcao joined him in the book for catching Trippier with his arm as he jumped for the ball.
Any hopes of a Burnley comeback were extinguished eight minutes from time when Arfield tripped Di Maria when he raced into the box.
Smalling did not put himself forward to try to score his first ever hat-trick
He left it to Van Persie, who clinically jabbed the ball past Heaton.
Source : PA
Source: PA