Carson saved from Derek Niven and Mark Allott in the shoot-out and then Hermann Hreidarsson drove the final spot-kick past Barry Roche to put the Londoners through.
But Iain Dowie's team had to fight all the way against a Chesterfield side who had already put out Manchester City and West Ham.
A third Premiership scalp looked a distinct possibility when Colin Larkin caught the visitors cold in the third minute to flick the ball in at the near post from a corner.
Charlton could have been two down seven minutes later when the the speedy Caleb Folan got away on the right but his attempted lob went harmlessly wide.
Chesterfield were giving Charlton no time to settle and Kevan Hurst went close with a 20-yard shot before the visitors finally tested the home defence.
Jimmy Floyd Hasselbaink almost turned in a low cross from Jerome Thomas in the 39th minute but 60 seconds later the Dutchman powered forward and unleashed a low shot from 25 yards that zipped past Roche.
But Charlton were stunned again at the start of the second half when an in-swinging corner from Alan O'Hare was headed in by Folan.
The home fans now sensed another shock was on the cards and Charlton responded by bringing on top-scorer Darren Bent for the disappointing Marcus Bent.
The move paid off in the 73rd minute when an Andy Reid corner was cleared only as far as Thomas whose low shot was touched in by Bent for his eighth goal of the season.
The tie went into extra time and Charlton pounced in the 93rd minute when Hasselbaink and Thomas combined before the winger sent in a low cross from the Chesterfield right which Hasselbaink smashed in from 12 yards.
Hasselbaink was denied by the outstretched leg of Roche five minutes later, but Charlton were only seconds away from victory when Amady Faye needlessly handled and from the free-kick veteran striker Wayne Allison was left unmarked to head past Carson from eight yards.
Chesterfield went into the shoot-out on a high and their hopes soared when Roche save Hasselbaink's penalty, but Carson responded by denying Niven and then Allott before Hreidarsson kept his nerve to put the Premiership side into the quarter-finals.