Pajtim Kasami hit the crossbar with a second-half spot-kick after Alex saw red, and Bryan Ruiz did the same in the subsequent shoot-out, with replays inconclusive as to whether the ball crossed the line.
Chelsea's win improved their miserable record of seven penalty shoot-out defeats from their previous eight and gave manager Andre Villas-Boas another chance to blood his youngsters in the next round.
Their starting line-up was a reflection of owner Roman Abramovich's desire to use the Carling Cup as a breeding ground for the club's youngsters.
They crashed out at the same stage last season after that policy was introduced and there were predictable signs of players getting to know each other early on in this game,
Only a last-gasp Stephen Kelly challenge on Florent Malouda prevented them taking the lead 16 minutes in however, the Frenchman also whipping in a cross two minutes later that Salomon Kalou glanced wide.
Kalou fired narrowly off-target after Romelu Lukaku had seen his shot saved from a lightning breakaway that highlighted the latter's strength and pace on what was his first Chelsea start.
Fulham almost scored out of nothing when Ruiz, who had a much better game than his disastrous debut 10 days earlier, brilliantly played debutant Orlando Sa clean through but Alex recovered to make a superb sliding tackle.
With half-time looming, Josh McEachran brilliantly burst into the box in what was his first appearance of the season and beat Fulham goalkeeper Mark Schwarzer, with Daniel Sturridge making sure as Kelly tried to clear only to be flagged offside.
Injury was added to insult as Sturridge was forced from the field and, before Frank Lampard could come on to replace him, Petr Cech was hurt colliding with Sa and failed to re-emerge for the second half.
Ross Turnbull came on and it got worse for Chelsea in an extraordinary opening two minutes after the restart.
Kalou fired into the side-netting after being released by Lampard and moments later Fulham won the penalty that saw Alex dismissed.
Ruiz's clever back-heel freed debutant Kerim Frei, who was brought down by Alex.
Referee Chris Foy deemed the Brazilian last man and produced a red card, but Kasami failed to capitalise, smashing the penalty against the crossbar.
Foy denied the visitors a second spot-kick when Oriol Romeu - who was having a mixed full debut - shoved over Sa before Villas-Boas threw on John Terry for McEachran.
Ruiz almost spoilt all his good work when he gifted the ball to Malouda 20 minutes from time but the winger shot straight at Mark Schwarzer.
Paulo Ferreira got a vital touch to Chris Baird's cross but then up-ended substitute Mousa Dembele for a free-kick which eventually found its way to the latter, only for Turnbull to save.
Cech's understudy also pulled off a brilliant reaction stop to deny Kasami but then flapped at the resultant corner, which Baird fired wide.
Chelsea were still attacking and Ryan Bertrand - another making his first Chelsea start - wasted a great one-two with Malouda when he shot straight at Schwarzer.
The drama reached fever pitch when Frei and Kasami cleared off the line from a corner before the former's finish sailed wide at the other end under pressure from Turnbull.
The marauding David Luiz almost snatched a late winner but finished weakly, all but ensuring extra-time.
The tireless Lukaku nearly had an early chance but lost a foot race with Schwarzer, Fulham substitute Bobby Zamora pounced on an error before shooting tamely at Turnbull, and Luiz blazed over when well placed.
Both sides had already made all their changes and Frei was booked for a tired tackle on Ferreira, while a sweeping move moments before half-time saw Malouda drill into the side-netting.
Chelsea continued to search for a winner after the restart, although Zamora also curled over from 20 yards.
Lampard failed to control a good ball from Kalou, who was then denied a tap-in by a brilliant interception from the outstanding Philippe Senderos.
Substitute Steve Sidwell almost volleyed a sublime winner against his former club, Luiz nodded a corner over the top and Lampard was booked before the shoot-out ensued.
Lampard's penalty was immediately saved by Schwarzer, with Zamora, Luiz, Sidwell and Terry all scoring before Turnbull's save from Dembele levelled matters.
Kalou, Baird and Malouda then netted before Ruiz smashed Fulham's fifth against the crossbar, and possibly over the line, but it was not given.