Wilfried Gnonto came off the bench to score the winner in the first minute of second half stoppage-time in the Welsh city.
It was a chaotic Sunday evening fixture that saw Swansea score the first of seven goals through Harry Darling after just eight minutes.
Leeds responded with a goal from Tottenham loanee Manor Solomon, but the hosts regained the lead courtesy of Liam Cullen's low strike.
An own-goal from Ben Cabango within ten minutes of the start of the second half levelled the scoring in the damp conditions.
The visitors went in front for the first time when Solomon netted his second only for Florian Bianchini to equalise at the death.
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However, it was Leeds who had the last laugh as substitute Gnonto found the back of the net just moments later to seal all three points.
Daniel Farke's side will now head into the new week sitting top of the Championship table with 32 points to their name.
They are ahead of Sunderland on goal difference after the Black Cats had to settle for a 1-1 draw away to Millwall the day before.
A delighted Farke told Sky Sports after the match: "It feels great [to be top], especially after such a game.
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"It's an advertisement for this league - I think Sky Sports will be pretty happy and also we are happy after a massive win.
"I didn't see that coming. I expected a tight game, never a 4-3 because it's so difficult to score goals against Swansea.
"But it was an excellent game for the supporters and for the neutrals as well. I think in the end we deserved to win because we won more or less every statistic.
"We were a bit sleepy in added time at the end of the first half and a bit sleepy in added time at the end of the second half, but in between we were excellent."
Sheffield United in third place also have a points tally of 32, but it's Leeds who lead the standings in the second-tier of English football.
The Whites are aiming to securing their return to the Premier League under boss Farke at the second time of asking this season.
Leeds looked set for automatic promotion in the previous campaign but four defeats in their last six games meant they finished third.
They overcame Norwich City in the play-off semi-finals, but a 1-0 loss to Southampton in the final ensured another Championship season.
Farke's side have only lost two league matches and are the second-highest scorers on 28 goals, one behind Middlesbrough.
However, with 30 games still left to play and only four goals separating first and third, promotion is far from a certainty for Leeds.
They return to Elland Road on Wednesday night to host Luton Town who are 16th in the table following their relegation last season.