The game looked to be heading towards a stalemate with four minutes remaining, but Yohan Cabaye converted from the spot before Fraizer Campbell came off the bench to seal a memorable win for the Eagles at the death.
Despite their barren league run, which has seen them go 12 outings without a victory, Alan Pardew's men have now won all four cup ties since the turn of the year and have at least one Wembley date to look forward to.
A tame opening quarter to the contest, in which neither side could muster a single shot on target, made way for a more lively affair once both teams found their rhythm.
Emmanuel Adebayor came closest to making a breakthrough in the quiet opening 25 minutes, seeing his headed attempt from seven yards out blocked aside on its way through to goal.
That came at a point at which Reading were receding more and more into their own half, but a rare burst forward down the right from Hal Robson-Kanu provided a corner which Michael Hector headed on target.
The ball was trickling its way through, though Wayne Hennessey - back in the starting lineup as the only change from last time out - got down in a routine manner to collect.
Just two minutes later Ali Al-Habsi made his first meaningful stop of the game, keeping out a curled Cabaye free kick that was well struck from all of 30 yards out.
It took until 10 minutes before the interval until Palace could carve open their lower-league opponents, as Mile Jedinak slotted a pass through for Adebayor to latch on to, only for his over-elaborate attempt to be thwarted by Al-Habsi.
Two more chances would follow deep in a cagey first half, the first of which saw Ola John fire into the side-netting from a tight angle with an open goal gaping, coming after Hennessey failed to collect a simple cross into the box.
Palace perhaps should have taken a lead into the interval, however, as Yannick Bolasie's saved attempt ended in the path of Joe Ledley positioned just six yards out, but the Welshman could only side-foot over the bar.
The Eagles were the side on top in the early stages of the second half, too, seeing one Jedinak effort kept out by Al-Habsi, while Scott Dann's header from the resulting corner was cleared off the line by Oliver Norwood.
That presented a chance for the competition's top scorer to come on, with Matej Vydra being restricted to just 30 minutes due to a recent hamstring injury.
The Watford loanee almost made an instant impression when playing in Stephen Quinn, who saw his shot saved by Hennessey, but only after some more Palace pressure up the other end that led to both Jedinak and Adebayor coming close.
Palace were continually being frustrated by Al-Habsi as the match entered its final quarter, this time after the stopper got across to his left to once more to deny Cabaye from a long-range free kick.
It could have been a different story for the Premier League outfit had John, who sneaked in to meet a back-post cross with his head 13 minutes from time, not seen his attempt blocked on the line by the leg of Hennessey.
Then came the big moment in the match five minutes from time when Jake Cooper put his arm across the chest of Bolasie, who was charging down a ball-to-nothing over the top, leading to Mike Dean pointing to the spot and reaching into his back pocket.
Al-Habsi got a hand to Cabaye's penalty, but it was not enough to deny the Frenchman firing a vital opener to put his side on the brink of a first Wembley semi-final appearance in more than two decades.
There was still time for Fraizer Campbell to come off the bench and smash home a second at the second time of asking right on the line, ensuring that the out-of-form Eagles will continue their cup adventure for at least another month.
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