Cardiff had dominated the majority of the 120 minutes, and won 1-0 on the night after Anthony Gardner, Palace's first-leg matchwinner, had put through his own net.
The hosts struck the woodwork three times through Kenny Miller, Filip Kiss and Aron Gunnarsson, while Palace played with 10 men from the 78th minute after captain Paddy McCarthy was dismissed for two fouls on the Scotland striker.
But the missed chances did not prove costly for Cardiff as Heaton, who is second choice behind David Marshall, saved from Jermaine Easter and Sean Scannell, before Jonathan Parr put the decisive spot-kick wide to hand the Welsh side a place in the Carling Cup final against Manchester City or Liverpool.
Cardiff took the lead in just the seventh minute as Gardner, who had given Palace their slender advantage, drew the Bluebirds level in the tie by putting through his own net.
Darcy Blake played Don Cowie in down the right and Gardner could only turn his cross past a stranded Julian Speroni.
Keeper Speroni had to be on his mettle again in the 11th minute as he beat away Peter Whittingham's fierce angled drive, with Anthony Gerrard narrowly failing to get to the loose ball.
Palace threatened little until Wilfried Zaha worked his way into the box, but Andrew Taylor got back to clear his attempted pass to Glenn Murray.
After such a frantic opening quarter the game eventually settled into some form of pattern, as Palace found a foothold in the contest.
But the hosts remained the dominant force, enjoying large swathes of possession and Miller should have doubled their lead after Parr's woeful header gifted Gunnarsson possession.
The midfielder played the Scotland striker in as Palace appealed in vain for offside, but his left-footed shot rolled agonisingly wide of the right-hand post.
And there was more anguish for Miller when he hammered a volley against the post moments before the interval.
Miller was again in the thick of the action on the resumption, picking his way past three defenders before working the ball to Whittingham, whose low shot beat Speroni but also the near post.
Palace had a rare attempt on goal after 52 minutes as Chris Martin turned well on the edge of the box after being found by Murray but fired over.
Whittingham showcased his set-piece prowess by curling in a free-kick that Speroni had to tip over the bar, with Miller's belligerence having forced a foul, which resulted in a booking, from McCarthy.
The Palace skipper was forced into a vital intervention moments later as Gerrard snuck round the back to head a Whittingham free-kick goalwards, scrambling the ball away to safety.
Cardiff were given a reminder of the threat the visitors posed on the counter as substitute Scannell's first contribution was to tee up Zaha, but the teenager could only blaze harmlessly wide.
Having missed that great opportunity to put themselves back in front on aggregate, Palace were then reduced to 10 men with 12 minutes to go as McCarthy was handed a second booking following another tussle with Miller, leaving the centre-half decidedly unimpressed by Howard Webb's decision.
Zaha redeemed himself by making a brilliant block in the final moments, getting back to slide in and deny Cowie and make sure the tie went to extra time.
With the extra man Cardiff dominated the entirety of the 30-minute period and would not have had to endure the agonies of penalties had Kiss not whacked a volley against the bar from inside the box, and Gunnarsson not done likewise by contriving to find the woodwork with his point-blank header.
Those misses looked like they might prove costly when Miller missed the first spot-kick, but Heaton saved low down from Easter and Scannell, before Parr failed to find the target to send the Welsh club to Wembley for the third time in five seasons and spark a minor pitch invasion by the delighted home support.