Andy Robinson scored with a ninth minute free-kick but Leeds were quickly back on level terms thanks to Jermaine Beckford, who netted from a narrow angle three minutes later.
Gary Monk headed home direct from a 23rd minute corner to restore the lead but when Ferrie Bodde was dismissed it looked as though Swansea's chance had gone.
Up stepped Jason Scotland in the 45th minute to pick up the ball on the half way line. He shrugged off two defenders before firing passed Casper Ankergren in the Visitor's goal.
Leeds were stung into action and had pulled one back within a minute of the restart thanks to skipper Alan Thompson who slighted a free-kick into the corner of the net with Swansea keeper Dorus de Vries slow to get across his line.
Swansea had made just one change from the side that beat Cheltenham on Boxing Day with fit-again Scotland replacing Guillem Bauza who was on the bench.
De Vries needed two attempts to grasp a Jonny Howson shot in the fourth minute but by then referee Andre Mariner had already blown for numerous fouls as tackles flew in hard from both sides.
Leeds fall back Andrew Hughes went in the book as early as the seventh minute and even at that stage it looked likely that one of the sides would end up with ten men.
The Swans were two-one ahead when Bodde received his marching orders for a stamp on Howson that led to the Leeds man being stretchered off with his leg in a splint.
The pivotal moment in the game came with Scotland's solo effort on the stroke of half time which gave his side valuable breathing space.
They still had to play the second half with only a one-goal advantage after Thompson had reduced the deficit but they defended magnificently and fully deserved their victory.