Swansea substitute Zan Vipotnik scored a dramatic equaliser in the sixth minute of second-half stoppage-time as they held Leeds to a 2-2 Championship draw at Elland Road.
The result sees Leeds sit in second place, two points behind league leaders Sheffield United and level on points with Burnley in third.
Willy Gnonto came off the bench to score a 86th-minute goal which looked set to give Daniel Farke's men victory - but Vipotnik fired home a late shot past Illan Meslier, who was arguably at fault for both Swansea goals, to level matters at the death.
United States midfielder Brenden Aaronson put Leeds ahead from close range inside just 35 seconds before Meslier saved a 13th-minute penalty from Josh Tymon.
After home goalkeeper Meslier dropped the ball from Tymon's corner in the 64th minute, Swansea's Harry Darling was on hand to fire home and make it 1-1.
Gnonto struck a fiercely-struck shot which flew into the net to put Leeds 2-1 ahead but the visitors kept probing and were rewarded late on.
Meslier's roller coaster afternoon
Tributes were paid to honour the memory of Leeds fans Christopher Loftus and Kevin Speight, who were killed in Istanbul before their team's UEFA Cup tie against Galatasaray in April 2000. A minute's applause took place before the game and in the 25th minute of the first half.
Leeds were boosted by the return of captain Ethan Ampadu after six weeks on the sidelines through injury.
They led almost immediately when full-back Jayden Bogle broke down the right flank and delivered a dangerous low cross.
Joel Piroe saw two close-range attempts at goal blocked by the visiting defence, but the ball fell kindly to Aaronson who fired the ball home from inside the six-yard box.
Swansea gradually steadied themselves and won a penalty when Rodon fouled Lewis O'Brien inside the 18-yard area.
Tymon stepped up to hit a low, left-footed effort which saw Meslier dive low to his left to make a fine save.
Swansea went even closer to an equaliser midway through the first half when Tymon's inswinging free-kick was greeted by captain Ben Cabango's glancing header.
His effort hit the inside of a post and rebounded before Leeds cleared the danger, but the hosts were far from their usual fluent selves.
After the break, the home side threatened through Wales winger Dan James, who hit a superb left-foot effort just wide in the 57th minute.
Farke then replaced Ampadu with Japan international Ao Tanaka and, moments later, Aaronson crossed from the right to Piroe, whose close-range volley was superbly repelled by Swans goalkeeper Lawrence Vigouroux.
Swansea then went close at the other end when a defensive clearance from a corner fell to Ronald, who hit a right-footed shot from 18 yards, which Meslier did well to push around a post for a corner.
From the subsequent inswinging delivery by Tymon on the right, Meslier dropped the ball and Darling fired home from inside the six-yard box.
Piroe found the space to hit a piledriver which Vigouroux did well to push away before Aaronson headed over a cross from James moment later.
Farke began to ring the changes and, after Pascal Struijk saw his header pushed out by Vigouroux, Gnonto was on hand to rifle home a fierce right-footed shot into the net from around 14 yards out.
But Swansea kept probing and when Joe Allen cleverly played in fellow substitute Vipotnik, the Slovenia international fired the ball home low and hard past Meslier to leave Elland Road stunned.
The managers
Leeds' Daniel Farke:
"I'm far away from punishing a player [in Meslier] because we win together, lose together and draw together. After such a game and such a heartbreaking finish, they don't need a manager who puts the result on the shoulder of players.
"It's a heartbreaking moment if you concede in this manner but it's football. If you want to take part in sport, you can't expect the sun to always be shining.
"We've had great winners in the last minute and the opponent had to suffer the disappointment we had today - Sunderland and Sheffield United felt exactly the disappointment we do. You have to suffer a little bit, that's normal, but then we go again.
"We are still in the situation that if we win our last games, we will finish in the top two. That's due to the fact they (Sheffield United and Burnley) still have to play each other but there's never a guarantee of success.
"Right now it's the final step which is always the most difficult. I was in this situation before and I know how tricky it is but I know how difficult it is to bring it over the line.
"If you ask me right now, I'm 100 per cent convinced we will play in the Premier League next season."
Swansea's Alan Sheehan:
"A point is the least we deserved - we were tactically excellent.
"We had a nightmare start and went 1-0 down but we kept going and it was a high-quality pass for the equaliser late on.
"Leeds are one of the best teams in the league - if not the best - and it's the toughest ground to go to.
"But Joe Allen played that pass into Vipotnik and it was wonderful to see the two substitutes combine.
"To come here and get a point shows the belief and the togetherness that we have right now."
