Andy Johnson fired the Eagles in front from the penalty spot on 79 minutes but Steve Coppell's Reading quickly responded in typical fashion to stretch their unbeaten run to 29 matches.
A lively first half saw good opportunities at both ends with Palace perhaps having the edge over the league leaders. The Eagles' best chance felt to Johnson after just three minutes.
The Palace ace raced clear after Reading messed up in midfield but Marcus Hahnemann produced a desperate lock to deny the Palace hitman.
Darren Ward was forced to make a crucial clearance before the ball was at the other end again, this time Johnson going close once more.
Reading then put the Eagles under some pressure as Dave Kitson stretched to put his shot over the bar, followed by a couple of further chances which he went close with - Kevin Doyle the provider.
Jon Macken and Johnson then combined to give Jobi McAnuff a half chance as the Palace midfielder curled over the upright.
And it got even better in the second half. Glen Little slipped through the Palace defence only for Gabor Kiraly to bend backwards and tip over the bar, while for Palace central defender Mark Hudson should have done better when he headed over an inch-perfect corner from Ben Watson.
The football continued to flow as Bobby Convey shot wide and then Little tested Kiraly once more. Watson and skipper Michael Hughes were outstanding for Palace, while Reading's Harper and Steven Sidwelll were an equal match.
The action was non-stop and the deadlock was eventually broken 11 minutes from the end. Fitz Hall managed to get to a 50-50 ball first and it fell kindly to Johnson who made his run.
Hahnemann left his goal only to clatter into the Palace striker. From the penalty Johnson powered his kick past the keeper and into the top right corner.
But within two minutes Reading were level as they responded with a superb move.
Little started it down the right, fed the ball inside to Doyle who in turn cleverly flicked the ball to Harper. The midfielder half-volleyed a ferocious shot from 15 yards out which gave Kiraly no chance as it soared into the top corner.
Watson could have won it for Palace in the closing minutes, but from McAnuff's cross he completely miskicked in front of goal.