The Welsh side will visit Wembley for the fourth time in five seasons as they saw off the brave but ultimately fruitless Eagles, who had Paddy McCarthy sent off in the second half
Malkay Mackay's men secured victory after Craig Conway, Rudy Gestede and Peter Whittingham scored from the spot. Jonathan Parr was the unlucky man for the Eagles as he blazed high and wide to ensure there will be a Welsh representation in the Wembley showpiece next month.
The Bluebirds, who trailed 1-0 from the first leg, were deserved winners on the night as they overcame a stern test from 10-man Palace, but were forced to go the distance after missing a string of chances in normal and extra-time.
They took the lead when Anthony Gardner, Palace's hero from the first leg, was unfortunate to divert Craig Conway's right-wing cross in at his near post via the woodwork.
Palace had Paddy McCarthy sent off for a second bookable offence with 13 minutes to go but Palace held on doggedly for around 45 minutes with some desperate defending to ensure that the semi-final would be decided from 12 yards.
The game had started at a frenetic pace, and even though Cardiff began to dominate possession, the Eagles were always a threat on the counter attack, spearheaded by the influential Wilfried Zaha.
Zaha, who has been linked with numerous Premier League sides, will have impressed his suitors after a brilliant performance as he terrorised his opponent Darcy Blake.
The 19-year-old sent over a teasing cross after turning Blake again with ease, but with the right-back and goalkeeper Heaton struggling, the ball was cleared off the line to the relief of Bluebirds fans.
Despite Zaha's efforts though, Cardiff were always in control thereafter and had a number of attempts to double their advantage but were rather wasteful in the final third.
First, Whittingham tested Julian Speroni from range with a stinging drive before Kenny Miller missed a gilt-edged chance when Aron Gunnarsson slipped in the Scotland striker who sent his effort just inches past Speroni's far post.
Miller then went one step closer to doubling Cardiff's advantage just before the break when he capitalised on slack defending from Conway's deep centre, before swivelling and firing a shot against the post.
Out of luck | Gardner turns the ball into his own net after first leg heroics
The Bluebirds continued their dominance in the second half as they forced several openings and began to carve the Crystal Palace defence apart at will.
Whittingham was at the centre of everything good about the home side's play and he went very close to ending stubborn Palace's resistance when he forced a stunning save from Speroni after an exquisite free-kick.
The second half was all about the former Aston Villa man as he teed up Anthony Gerrard with a sumptuous chip, but the defender's effort was hacked clear by McCarthy.
Whittingham and Miller wasted further chances before Palace created their first real chance of the half when substitute Sean Scannell freed Zaha, who blazed over when well-placed.
The game changed with 13 minutes left, when McCarthy was dismissed for coming through the back of Miller, who used his experience and nous to win the foul.
Cardiff came close in the final stages as Zaha's block denied Don Cowie a clear sight at goal and Conway struck just inches wide, but the game would go to extra-time.
Crystal Palace defended manfully throughout the extra-time period as they dealt with the added threat of Gestede, who was introduced from the bench to try and use the extra man to their advantage.
Cardiff mustered two clear-cut chances as first Cowie sliced over, before substitute Filip Kiss cracked the crossbar with a rasping drive.
The frenetic pace of the game petered out as both sides tired but it was the Bluebirds who continued to press and should have scored in the final few seconds when Gunnarsson somehow contrived to head against the bar from just a few yards out.
It appeared that Cardiff's profligacy may come back to haunt them in the shootout, but despite Miller's miss, Palace were poor from the spot as Easter and Scannell had their penalties saved comfortably by Heaton.
So it was left to poor Parr to keep his side in it, but it was not to be for Palace, who will now concentrate on league matters, while Cardiff await Manchester City or Liverpool.