Sam Greenwood scored a first-half penalty for the visitors before a delightful Kayky chip brought us level just after half-time.
Romeo Lavia was then sent off for City for a second bookable offence, before Greenwood grabbed his second for the Yorkshire side.
Brian Barry-Murphy's side were faced with a tough test at the Academy Stadium, with Leeds naming first-team stars Luke Ayling and Patrick Bamford in their starting eleven as they returned from injury.
But despite this, the EDS controlled proceedings right from the first whistle in the bitterly cold conditions.
Palmer looked particularly dangerous on his return to the side, hitting the crossbar from outside the area with just five minutes played, before firing over Luke Bolton's cutback from close range a few minutes later.
The visitors were then awarded a penalty after Crysencio Summerville was bundled over in the City area, with Leeds captain Greenwood converting past Mikki van Sas.
Going into the break behind, Barry-Murphy would have been looking for an instant reaction to get us back into the game, and that is exactly what he got.
City drew level just two minutes into the second half, when Kayky brilliantly broke away from the defender inside the area to meet Bobb's assist before delightfully dinking the ball over the onrushing Klaesson to level the scores.
We kept pushing forward in search of a second, with Palmer and McAtee seeing efforts fly wide of the Leeds goal.
But the game then turned on its head in the space of a minute, as Romeo Lavia was sent off after receiving a second yellow card for a challenge on Nohan Kenneh, before Greenwood converted from inside the area just seconds later, to restore Leeds' advantage.
However, City didn't let their heads drop, and managed to bring themselves back level when Palmer and McAtee linked up brilliantly, with the latter spinning away from his defender inside the area before firing a powerful strike under Klaesson.
We kept pressing forward despite being a man down, and were rewarded when Palmer squirmed through a crowd of bodies in the 90th minute, before firing home after his initial effort bounced off the Leeds keeper, to secure three crucial points.
HOW WE LINED UP
WHAT IT MEANS
The EDS remain in fourth in the Premier League 2, but move level on points with Tottenham thanks to Palmer's late winner.
It was our first league win since October, and ended a four match winless run in the league for Barry-Murphy's side.
The EDS are back in Premier League 2 action this weekend, with a tricky away test at third-place Tottenham Hotspur in the offing.
Kick-off is at 12:00 (UK) on Saturday, 4 December.
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