Shunning their 15 point penalty, they overcame Swansea's strong defence and second-half goals by Jermaine Beckford and David Prutton saw them home
Still without their captain Alan Thompson, Leeds were unchanged, while Swansea replaced injured wingers Leon Britton and Andy Robinson with Thomas Butler and Paul Anderson.
Dutch midfielder Ferrie Bodde also returned after suspension.
Without a win at Leeds for 58 years, Swansea arrived on the back of successive victories over Carlisle and Cheltenham but they nearly fell behind after 20 minutes of pressure.
Jamie Clapham's corner was met by Rui Marques whose header seemed destined for the net until Dorus de Vries pulled off an excellent diving save.
Prutton swung his boot at fresh air after a glorious solo run into the Swansea penalty box and the Welshmen narrowly survived again when Tresor Kandol's 15-yard shot from the left grazed the far post.
Swansea were rarely seen as an attacking force, with strikers Jason Scotland and Darryl Duffy starved of opportunities.
And the Welshmen were again fortunate to remain on level terms when, two minutes into the second half, Beckford forced a finger-tip save from the alert de Vries.
Two goals in five minutes broke Swansea's resistance.
First Beckford turned Seb Carole's free-kick past de Vries on 62 minutes for his sixth goal of the season.
Then Prutton beat the keeper with a stunning 25-yard drive to register his first goal for the club.
Butler proved a handful for the home defence on the right flank but there was no stopping Dennis Wise's team who are now chasing an outright club record of eight wins.