Villa had been stunned when Darren Ambrose's spot-kick cancelled out Gabriel Agbonlahor's 14th goal of the campaign.
But Carew had the final say for the Carling Cup finalists, with his double blast after both penalties were conceded by Matthew Lawrence.
It put Villa through to their first FA Cup quarter-final in 10 years, their first last-eight appearance since they reached the 2000 final.
It was the perfect tonic for Martin O'Neill's side before their Carling Cup meeting with Manchester United on Sunday.
Villa began strongly and low cross from James Milner found its way through to Agbonlahor but he was unable to make a clean contact with his shot and Palace goalkeeper Julian Speroni was able to collect.
Carew then made good contact with a volley from Ashley Young's centre but it flew straight at Speroni from a narrow angle.
Agbonlahor created a chance for himself when turning and shaking off the challenge of Lawrence but Speroni was able to beat out his left-footed drive.
Speroni had to be alert to tip over a rising drive from an acute angle by Ashley Young, who was causing plenty of problems to the Palace defence.
Carew then had a powerful shot turned aside by Speroni as the home side upped their game.
The Palace goalkeeper was relieved when a fierce attempt by Ashley Young drifted wide after it taken a strong deflection and left him stranded.
Stephen Warnock wasted a good chance after being freed by Milner inside the box and, with Carew completely unmarked, he over-hit his pass across the danger area.
Palace enjoyed a rare spell of pressure and Nick Carle had a low effort blocked from the second of two Ambrose corners in quick succession.
Speroni was perfectly placed to hang onto a shot on the turn from Milner as Villa quickly regained the initiative.
And their pressure finally paid off when Agbonlahor broke the deadlock three minutes before half-time.
There was again controversy with the ball appearing to go out of play off Milner but referee Martin Atkinson decided it had struck Carle.
Agbonlahor was first to react to Ashley Young's centre and beat Speroni with a back header into the corner of the net.
The decision may provoke Neil Warnock's ire after Palace were denied victory in the first game when Villa captain Stiliyan Petrov scored a late equaliser from a corner which should not have been awarded.
Lawrence became the first player to be booked after 51 minutes for a challenge on Ashley Young before Villa goalkeeper Brad Guzan comfortably dealt with Ambrose's 20-yard effort in a Palace raid.
Nathaniel Clyne then followed Lawrence into Atkinson's notebook for bringing down Agbonlahor.
The England forward had the chance to double Villa's lead when racing onto a long ball from Luke Young but his shot lacked conviction and did not seriously extend Speroni.
The Argentine produced an excellent save low to his right to keep out a low shot from Ashley Young after a powerful run.
But Palace were now more in the game and a cross-cum-shot from Neil Danns was only just wide of the mark.
Palace got back on level terms after 72 minutes when Stephen Warnock brought down Lee in the penalty area and Ambrose converted the subsequent penalty.
But Villa retaliated and Carew earned - and converted - two more penalties to send Villa through to the last eight.
The Norway international was brought down by Lawrence after 79 minutes and made no mistake himself from 12 yards.
Then, with two minutes remaining, Lawrence was again found guilty of fouling Carew inside the box - and the striker again tucked away the penalty.