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.
After early Villa pressure, 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 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.
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.
ASTON VILLA: Guzan, L Young, Dunne, Cuellar, Warnock, Downing, A Young, Milner, Delph (Sidwell 76), Carew, Agbonlahor
CRYSTAL PALACE: Speroni, Clyne, Lawrence, Butterfield, Ertl, Davis, Derry, Ambrose, Danns (Djilali 90), Carle (Comley, 83), Lee (Andrew 90)