Jose Mourinho's side were facing a defeat at Old Trafford for only the second time this year thanks to first-half goals from Ashley Barnes and Steven Defour, before Jesse Lingard came off the bench to earn his side a point.
It means United have still only lost to Manchester City at home in 2017, but they will be 15 points behind the runaway Premier League leaders if Pep Guardiola's side beat Newcastle United on Wednesday.
The Red Devils endured a tough week before Christmas as they exited the EFL Cup to Bristol City before conceding a last-minute equaliser in a 2-2 league draw with Leicester City, and they found themselves behind after only three minutes when Barnes capitalised on their failure to defend a free-kick.
United pushed for an equaliser, but were dealt a sucker punch when Defour, once linked with a move to the club, struck a superb free-kick before the interval.
Lingard came off the bench to flick home and give his side a lifeline, and Burnley were denied a first away league win over United since September 1962 when the England international fired home in second-half injury time.
The draw sees Burnley stay in seventh place and keeps United in second, one point above Chelsea, but the result has surely struck a hammer blow to their hopes of hauling in their neighbours at the summit.
2 - Jesse Lingard is the first Man Utd substitute to score a Premier League brace since Chris Smalling did so in February 2015 (also v Burnley). Impact.— OptaJoe (@OptaJoe) December 26, 2017
Mourinho had branded his players "childish" after their collapse against Leicester and the way they started against Burnley would have done little to change his view.
Marcos Rojo conceded a free-kick on the edge of the penalty area in only the third minute, United's defence failed to deal with the delivery and Barnes was given an easy task of drilling home from inside the six-yard box after the ball fell kindly at his feet.
The home defence looked in disarray in the early stages and Scott Arfield clipped the top of the crossbar with a volley from Johann Berg Gudmundsson's cross, after United again failed to clear their lines.
The hosts recovered from their early jitters and began to pin Burnley back, with Luke Shaw - handed a start at left-back in a league game for only the second time this season - stinging Nick Pope's hands with a strike before Paul Pogba's header was clawed away from the top corner.
But Burnley stunned Old Trafford for a second time nine minutes before the break, when Defour curled a sublime free-kick into the top-left corner from 25 yards out to double the visitors' lead.
United could easily have snatched at least one goal back before the break, with Ben Mee clearing brilliantly off the line after Marcus Rashford had beaten Pope, with Pogba and Zlatan Ibrahimovic also coming close.
Mourinho reacted at the break by bringing on Lingard and Henrikh Mkhitaryan for the ineffective Ibrahimovic and Rojo, and the England man missed a glorious chance to snatch a goal back moments after entering the fray, as he turned the ball off Pope and onto the crossbar when unmarked barely three yards from goal.
Yet with 53 minutes gone, Lingard made the most of his second chance, steering Ashley Young's low cross into the bottom-left corner with a sublime backheel that just eluded Pope's fingertips.
United remained camped in the Burnley half as the home fans tried to keep hope alive, with Mkhitaryan heading over the bar and Pogba firing a free-kick just too high, and the pressure finally told in the first minute of injury time when Lingard arrowed a low strike through the crowded box and into the bottom-right corner after the visitors, for once, failed to clear.
The draw is another commendable result for Burnley on the road this season, with Sean Dyche's side having beaten Chelsea and Everton and taken points away at Liverpool and Tottenham, but they – just like United – are unlikely to celebrate with much gusto.
Key Opta stats:
- This was the 20th time a Jose Mourinho managed side had been two-goals down in a Premier League game, but only the second time they hadn’t lost (W0 D2 L18).
- The Clarets became just the second different team to score twice against Jose Mourinho's Man Utd at Old Trafford, after Man City (twice, in September 2016 and earlier this month).
- Jesse Lingard has had a hand in 10 Premier League goals this season (6 goals, 4 assists in 16 apps) – as many as in his previous three seasons combined (5 goals, 3 assists in 51 games).
- Manchester United have had 61 shots across their last two home games against Burnley, but have scored just two goals and picked up two points.