Clarets pay the penalty in Carabao Cup
Stand-in Burnley skipper James Tarkowski was the only player who failed to convert from 12 yards out as goalkeeper Andy Lonergan guessed the right way.
The third round tie went to penalties after the game finished 2-2 after extra-time, with all four goals arriving in an enthralling final 10 minutes in normal time.
And they all came courtesy of the substitutes with Hadi Sacko, Chris Wood, Pablo Hernandez and Robbie Brady all getting on the scoresheet.
Burnley made seven changes for the fixture against the Championship leaders yet controlled the tie without having to move out of second gear.
The performance was a shining example of the Clarets' squad depth as the tie turned in to a training session for the hosts on occasions.
Thomas Christiansen had carried out surgery on his Leeds United side going in to the clash at Turf Moor, making nine changes from the side that lost out at Millwall at the weekend, and they couldn't cope with Burnley's class.
However, the first half ended goalless as the home side failed to score the goal that their first half showing had warranted.
Scott Arfield, the scorer against Liverpool at Anfield, almost found the back of the net again when collecting Sam Vokes's knock down but the midfielder was denied by Lonergan on this occasion.
The Whites rarely threatened but even when they did craft an opening the visitors proved far from convincing in front of goal.
Jay-Roy Grot had done the hard work when turning skipper Tarkowski to create room for himself from Stuart Dallas's pass but the Dutch teenager could only roll the ball in to Nick Pope.
Phil Bardsley, who stepped in for Matt Lowton at right back, almost capped an impressive first half display with the opener but Lonergan watched as his daisy-cutter from 20 yards out skidded past the post.
The Clarets were working beautifully between the lines, pulling the away side out of shape at every given opportunity, and former United full back Charlie Taylor was one man to exploit that on more than one occasion.
After getting down the left hand side, behind Gaetano Berardi, the 24-year-old whipped in a fabulous delivery which Ashley Barnes powered wide with his head.
United's Swiss full back was caught out again moments later as Johann Berg Gudmundsson's set-piece dropped over his head but Kevin Long, with the goal at his mercy, headed wide at full stretch.
More opportunities followed but the Premier League outfit couldn't quite hit the spot.
Arfield side-footed wide from six yards out having run on to Vokes's squared pass while Barnes flicked his header just wide of the upright when beating Luke Ayling to Bardsley's cross.
A similar pattern emerged after the interval as Gudmundsson's curling effort had bums off seats but the winger's curling effort whistled past the post after evading the fingertips of Lonergan.
Barnes flicked another header past the post after out-jumping his marker to meet Gudmundsson's centre while, at the other end, substitute Sacko was reduced to an audacious dip from distance but the ball veered well off target.
The hosts had done everything right in the match apart from score but they were getting closer and closer.
Having stretched their opponents once more, Gudmundsson fed Bardsley in to the penalty area and the right back's cross-cum-shot crept past the woodwork.
Burnley had been subjected to a barrage of shots against them in recent outings in the top flight - a total of 86 to be precise from games against Spurs, Crystal Palace and Liverpool - but the shoe was on the other foot in the Carabao Cup.
Burnley, though, for all their dominance, hadn't really tested Lonergan but the former Preston North End stopper stayed alert to beat Ashley Westwood's drive to safety.
There only looked like there was going to be one winner, until United's substitutes combined with the most undeserving of sucker punches.
Hernandez, with his first touch of the evening, found Sacko in acres of space and the French winger closed in on goal before firing between Pope's legs with 10 minutes remaining.
Hernandez almost added insult to injury from a free kick in a dangerous position on the edge of the box but the one-time Valencia midfielder's attempt dropped on to the roofing of Pope's goal.
After Lonergan spread himself well to deny Gudmundsson at the back post, with the Icelandic international pouncing on Tarkowski's header, the Clarets were heading for the competition's fire exit.
That was until substitute Wood intervened in the last minute of the game to haunt his former employers.
The striker, who was United's leading scorer with 27 goals last term, levelled from the spot, sending Lonergan the wrong way after Berardi had impeded Long.
The majority inside the stadium thought that was it but there was still time for a huge twist in the tale as the fourth official signalled for six minutes of time added on.
Burnley pressed for the winner but, again, it was Leeds who had one leg in the fourth round when they were awarded a penalty.
Darren Bond's decision seemed harsh as the referee penalised Tarkowski for a shirt pull on Kemar Roofe. Hernandez, however, didn't care and the Spaniard stepped up to confidently stroke the ball home to restore United's advantage.
While the travelling fans celebrated, hoards of home supporters marched out of the stadium in frustration and disappointment. What a terrible decision that turned out to be!
With seconds remaining the stage was set for Brady after Wood was upended just outside the penalty area.
The Republic of Ireland international stood over the ball, took a few steps back, gauged the distance, before floating a wonderfully weighted left-footed shot over the wall and out of the reach of Lonergan to force extra-time.
With neither side able to make their mark in the additional 30-minute period, the tie was to be decided by the lottery of penalties.
Pierre-Michel Lasogga drew first blood for Leeds with Hernandez and Mateusz Klich following suit.
Wood, Barnes and Brady responded for the Clarets to level after three spot kicks apiece but Tarkowski's miss handed Leeds the initiative.
And the Championship side capitalised as Ezgjan Alioski netted before Dallas scored the decisive penalty.