For the second season running the Clarets were denied an historic Old Trafford victory by an injury-time equaliser as they were only moments away from handing Ole Gunnar Solskjaer a first defeat as Manchester United’s interim boss.
Sean Dyche’s men did prevent Solksjaer from continuing his perfect start and becoming the first manager in the Premier League era to start with seven successive victories.
But it could have been even better for a Burnley side who were again within touching distance of a first win at United since 1962 as they drew here for the third time in a row.
Just like last season the Clarets led 2-0 and this time they held the advantage to the 87th-minute thanks to second-half strikes from frontmen Ashley Barnes and Chris Wood.
Dyche’s warriors battled magnificently to earn and then protect their lead in a magnificent performance.
But a Paul Pogba penalty gave United late hope.
And with Solksjaer’s appointment bringing back memories of the Sir Alex Ferguson era, they then struck in Fergie time through Victor Lindelof to cruelly deny the Clarets at the death.
It was a devasting blow for a Burnley team that had given everything but once again Dyche’s side showed beyond doubt their desire to win their relegation fight as they put another priceless point on the board and extended their own unbeaten run to five league games.
With eight straight wins in all competitions behind them, it was perhaps inevitable that United would look to maintain that momentum with a lightning start.
And the Clarets had to come through a testing opening 10 minutes before getting a foothold in the game.
They were grateful to Charlie Taylor for two alert pieces of defending – including a vital block to get in the way of a shot from Marcus Rashford.
But the Clarets were also glad to see a glaring miss from Rashford after the England man had swapped passes with Romelu Lukaku before poking the ball wide.
Phil Bardsley, passed fit to face his former club, mirrored his full-back partner with an excellent covering tackle to snuff out a threat from Juan Mata.
But the Clarets, reverting to the starting line-up that had kept the league run going ahead of their FA Cup exit at Manchester City last weekend, never let United have it all their own way.
And for a spell mid-way through a goalless first half they forced the home side onto the back foot as the terrier-like Ashley Westwood led a pressing game that threatened to open a route to goal.
Barnes also never gave the United back line a moment’s peace and following a typical burst from Westwood, it took a crucial intervention from Phil Jones to take the ball off Barnes’ toe as he prepared to shoot.
Barnes wouldn’t be denied for long, though, and six minutes into the second half he struck to give the Clarets the lead.
Jack Cork did the spadework as he robbed Andreas Pereira and slipped a pass to Barnes who, like last season here, opened the scoring with a thunderous left-footed shot past David De Gea.
Not surprisingly there was an instant response from the home side, but former United keeper Tom Heaton, enjoying another top-class return to his old stamping ground, turned away a snap-shot from Rashford.
And then another player who had started out at United almost had his moment of glory as Clarets’ teenager Dwight McNeil, who again looked at home on the biggest of stages, stung the hands of De Gea at the other end with a rasping drive from all of 30 yards.
But most of the attacking was coming Burnley’s way as United poured men forward in search of an equaliser and sent on Alexis Sanchez and Jesse Lingard’s whose two goals on Boxing Day last year had dashed Burnley’s hopes of ending that long wait for an Old Trafford win.
Solksjaer’s men peppered the penalty area but the Clarets refused to budge with James Tarkowski seemingly everywhere alongside the colossal Ben Mee as the back four to a man performed heroics in front of an inspired Heaton, who made a fine save to keep out a near-post flick from Lukaku.
And their collective efforts were rewarded when Wood remarkably made it 2-0 with nine minutes to go.
The fit-again Johann Berg Gudmundsson had replaced a tiring McNeil and when he teed up Westwood, the midfielder’s cross was coolly nodded in by Burnley's leading scorer at the far post.
The Clarets’ superb away following celebrated wildly but when United pulled one back from the penalty spot with three minutes of normal time to go those supporters were left biting their nails.
Jeff Hendrick’s tug on Lingard was penalised by referee Jon Moss and Pogba converted from the spot to spark the comeback.
With five minutes added at the end there was still time for United to deny the Clarets again and, agonisingly, they did so.
Heaton - having just turned away another Pogba effort in impressive style - made a magnificent save to keep out a header from Sanchez but the unlikely figure of defender Lindelof was there to ram home the loose ball and break Burnley hearts with his first goal in English football.
Manchester United: De Gea; Young, Jones, Lindelof, Shaw; Pereira (Lingard 62), Matic, Pogba, Rashford, Lukaku (Sanchez 68), Mata. Substitutes: Romero, Smalling, Fred, Dalot, Herrera.
Burnley: Heaton; Bardsley, Tarkowski, Mee, Taylor; Hendrick, Westwood, Cork, McNeil (Gudmundsson 77); Barnes, Wood. Substitutes: Hart, Vokes, Gibson, Ward, Vydra, Benson.
Booked: Barnes, Taylor
Referee: Jonathan Moss