Steven Defour and Ashley Barnes scored memorable goals before the Clarets were denied a first win at Old Trafford since 1962 by Jesse Lingard’s injury-time equaliser.
Barnes created his little bit of history and Defour struck a stunning free-kick as Sean Dyche’s men led 2-0 at the break as they went in search of only a second win in 24 games against United.
A second-half double from substitute Lingard dashed those hopes in heart-breaking fashion as United only just extended an unbeaten home record on Boxing Day stretching back to 1978.
It also meant Sean Dyche’s men had gone three games without a win in the Premier League for the first time this season.
But this was still another big point for the Clarets as they passed a major test of character after losing 3-0 at home to Tottenham three days previously to stay within a point of the top six and show that they are in it for the long haul.
The Clarets were attempting to re-write recent history and made the perfect start in their quest to make it four Boxing Days wins over the 20-times champions of England.
Burnley hadn’t led at Old Trafford since 1975 when Ray Hankin struck first in a 2-1 defeat.
But just three minutes in, the impressive Barnes bridged a 42-year gap to put the Clarets 1-0 up.
The striker, who replaced the injured Chris Wood in the only change to the Burnley side beaten by Tottenham, gleefully rammed home his third goal of the season from close range after Johann Berg Gudmundsson’s perfectly flighted free-kick – which earned Marcos Rojo a booking for a foul on the fast-starting Jeff Hendrick – had dropped in the six-yard box.
And in a fantastic start from the visitors, the lead was almost doubled 10 minutes later.
A superb build-up ended with Gudmundsson whipping in another dangerous delivery which Scott Arfield clipped onto the top of the crossbar with a wonderful cushioned volley.
The Clarets’ fans were in dreamland but inevitably there was a response from a United side beaten only once at home in 42 games.
Jose Mourinho had certainly not scrimped on his striking resources, although the hosts’ first shot on target came from left-back Luke Shaw, whose stinging drive was pushed away by Nick Pope.
Pope also kept a deflected Paul Pogba header out of the top corner and some heroic defending in front of him prevented United from finding a way through as a piece of individual brilliance from Defour saw the Clarets go into the break 2-0 in front.
Ashley Young’s trip on Scott Arfield more than 30 yards from goal gave Defour the chance to line up a free-kick and the Belgium international – once coveted by United – curled an unstoppable effort past Dave De Gea and into the top corner to score his first league goal in 15 months.
United tried to hit back straight away but Ben Mee was perfectly positioned to clear off the line from Marcus Rashford, matching the superb defending he and Kevin Long had shown earlier to block attempts from Juan Mata and Zlatan Ibrahimovic.
The legendary Swede, who had been similarly frustrated in last season’s 0-0 draw which earned Burnley their first point at Old Trafford since 1974, was then replaced at half time as Mourinho made a double change.
And it was his replacement, Lingard, who got United back into the game with a goal eight minutes after the re-start.
The England international had just been denied by an incredible point-blank from Pope when he was picked out again by another low cross from Young, which he cleverly flicked beyond Pope with his right instep.
United, with fellow substitute Henrikh Mkhitaryan seemingly on a mission to prove a point, then camped in the Clarets’ half as they hammered away at the Stretford End to try and force an equaliser.
But Burnley refused to buckle again until the 91st-minute as they defended heroically and Dyche sent on Sam Vokes and then Jon Walters to provide an out-ball and an added presence further up the pitch.
Despite their pressure and domination of possession, the home side weren’t allowed a clear sight of goal until assistant referee Lee Mason had indicated five minutes of stoppage time and Pogba floated a free-kick into the Burnley box.
The ball was half cleared but fell for Lingard and through a crowd of bodies he was able to lash a shot into the back of the net to cruelly deny the Clarets a famous win, but not a point to go with those earned at Chelsea, Tottenham and Liverpool as Dyche's men again proved they can mix it with the best.
Burnley: Pope, Bardsley, Long, Mee, Taylor, Gudmundsson, Cork, Defour (Vokes 67), Arfield, Hendrick, Barnes (Walters 80). Substitutes: Lindegaard, Lowton, Marney, Westwood, Wells.
Booked: Barnes, Taylor, Gudmundsson, Bardsley, Pope, Vokes, Cork
Manchester United: De Gea, Young, Jones, Rojo (Mkhitaryan, ht), Shaw, Pogba, Matic, Mata, Rashford, Ibrahimovic (Lingard ht), Lukaku. Substitutes: Romero, Lindelof, Blind, Herrera, Tuanzebe.
Booked: Rojo, Shaw, Mkhitaryan
Referee: Martin Atkinson