Richard Cresswell's return from a long injury absence, when he left the bench in the 72nd minute, was a bonus for Leeds boss Dennis Wise.
Southend's 5-0 trouncing of QPR on Friday night meant Leeds were destined to remain at the foot of the table regardless, but Palace haven't given up hope of reaching the play-offs.
Leeds have lost 19 Championship games this season and kept a clean sheet in only one of their last 14 but the three points were greeted with wild celebrations by their fans.
The defeat was a blow to Palace's hopes of qualifying for the play-offs but' after impressing early on, they went into their shell and paid the price for early missed chances.
Leeds, whose latest blow was to lose Alan Thompson to a training injury, should have fallen behind in the 14th minute when Carl Fletcher just failed to connect with Mark Kennedy's low left-footed cross.
Fletcher then immediately intercepted a back-pass but goalkeeper Casper Ankergren, on loan from Brondby, marked his debut with an excellent block.
Five minutes later the Dane allowed Danny Butterfield's centre to slip from his grasp but Palace had no-one in position to take advantage.
Palace were dominating possession and Leeds' central defenders Rui Marques and Heath failed to stop a determined gallop by Jamie Scowcroft, only for Fletcher to shoot straight at Ankergren.
However, it was defender Heath who headed Leeds into a 27th minute lead against the run of play. Leeds won a corner on the left, and when Blake's inswinger bounced in the goalmouth, keeper Gabor Kiraly failed to move to the ball and Heath headed in unchallenged.
It was dreadful defending by Palace who had looked by far the more fluent side up to that point. Leeds' goal gave them a confidence boost and Butterfield had to make a goal-line clearance from Jonathon Douglas.
Then Kiraly atoned for his earlier error by showing razor-sharp reactions to beat out Ian Moore's header from another menacing corner delivered by Blake.
Cresswell's introduction paid off when he made a glorious run before putting Blake through to score.
But Leeds couldn't relax and Kennedy's free-kick was headed in by Cort to set up a nerve-jangling finale.