The visitors appeared to be heading for victory when Ezgjan Alioski put then in front with 10 minutes to go but Palmer pounced in a crowded area to score his first goal for the club.
Pierre-Michel Lasogga had put Leeds in front in the first half before Andreas Weimann punished a defensive mistake just before half-time and the late drama means Derby have now won only once in six games.
Leeds had struggled in recent seasons at Derby but they started brightly with Pablo Hernandez connecting with a volley from the edge of the area that was blocked by Richard Keogh.
David Nugent almost got in for the Rams before a mistake by Laurens De Bock gave the striker another sight of goal – but Liam Cooper cleared.
The home side went close again in the 16th minute. An inswinging corner from Tom Lawrence fizzed across the six-yard box and Matej Vydra appeared to get a touch as it went just wide of Felix Wiedwald’s left post.
Leeds were coming under increasing pressure and they had another escape in the 21st minute when Vydra caught Pontus Jansson in possession and played in Tom Lawrence, whose cross was headed wide by a diving Nugent.
Derby were hustling Leeds into mistakes and they were opened up in the 32nd minute when a flighted pass from Tom Huddlestone put Andreas Weimann in behind but his lob from just inside the area landed on the roof of the net.
Leeds had hardly threatened but two minutes later they were ahead when a crossfield pass sent Stuart Dallas in on the right and his cross was headed in from close range by Lasogga.
Derby almost equalised when Lawrence struck the bar from 20 yards but they levelled in first half stoppage time when Cooper could only head Huddlestone’s long ball behind him and Weimann scored from eight yards.
The home side had the first chance of the second half when Nugent headed Lawrence’s cross straight at Wiedwald in the 53rd minute and Nugent fired wide eight minutes later from Weimann’s cross.
Paul Heckingbottom’s men were losing the ball too easily and another slip led to Lawrence driving in a low shot which Wieldwald was unable to hold – but they had a chance in the 71st minute when Cooper headed over a Dallas cross.
Derby were caught in the 79th minute when a free-kick was cleared and Samuel Saiz broke before playing in fellow sub Alioski, who shot against Scott Carson but headed in the rebound.
However, Palmer had the last word when Leeds failed to clear and the on-loan Chelsea forward fired in from close range to salvage a point.
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