After Chelsea dominated a goalless first-half, Stoke led when Jon Walters was allowed to turn on the edge of the box before firing in off the crossbar.
Loic Remy drilled in an injury-time equaliser, seconds before Stoke's Phil Bardsley was sent off for two bookings.
Stoke scored all of their spot-kicks, Eden Hazard missing the final penalty.
Potters keeper Jack Butland was the hero for the home side, saving the Belgian's high effort to put Stoke into the quarter-finals.
'Sacked in the morning
Stoke fans took great pleasure in goading Mourinho in the wake of Walters' goal and again after Butland's penalty save, suggesting that the Chelsea manager might be looking for alternative employment on Wednesday morning.
Several national newspapers have suggested that the Portuguese boss could be sacked if the Blues lose to Liverpool at the weekend, while the Daily Mirror claimed senior players are expecting his departure in the near future.
And as the clock ticked past the 90-minute mark, the pressure on Mourinho looked set to increase.
The France forward, who replaced the injured Diego Costa in the first-half, fired into the roof of the net after Willian's corner was flicked on by Kurt Zouma.
The Blues looked the likelier to win the game in extra-time, Willian shooting wide and Butland denying young Brazilian defender Kenedy.
But, after nine faultless penalties, they were knocked out of the competition by Butland's brilliant save.
Chelsea show unity in defeatDespite the Premier League champions' dreadful start to the season, any suggestions of disharmony in the Blues ranks looked wide of the mark at Britannia Stadium.
The visitors showed no signs of their recent troubles in the first half, passing the ball around confidently and stretching the Stoke back line with their sharp movement.
England international Butland thwarted the visitors with a trio of top-class saves in the opening half-hour, twice denying Diego Costa and then Oscar, while Chelsea skipper John Terry's cheeky flick from Willian's cross was ruled out for offside.
And the character showed to level deep in stoppage-time suggested that the players were prepared to fight for Mourinho.
History stays on Stoke's sideStoke started the match having never lost to Chelsea in the League Cup, winning four and drawing three of their seven previous ties, including victory in the 1972 final.
And the home fans thought their side had done enough to extend that record within 90 minutes.
The Potters were left deflated by a dramatic turnaround in the dying moments of normal time and were forced to contemplate playing another 30 minutes with 10 men.
But they showed resilience throughout extra-time, keeping the visitors at bay - even with forward Mame Diouf forced into an emergency full-back role - to force the tie into a shootout.
Now Mark Hughes and his men can look forward to a second League Cup quarter-final in three seasons as the Potters aim to win the trophy for the second time.
Shawcross shores up Potters defenceStoke welcomed back Ryan Shawcross for his first appearance of the season after the skipper had back surgery in the summer.
And his return was perfect timing for Potters boss Hughes, who watched his team produced a shoddy defensive display in the 2-0 Premier League defeat by Watford on Saturday.
The once-capped England international impressed for the home side, marshalling his defence superbly - particularly after the break - to keep out Chelsea until Remy's late intervention.
Manager reactionStoke boss Mark Hughes: "Full credit to the lads because they showed tremendous character and determination and nobody can ever level that we don't have that in abundance inside our dressing room.
"It was there for everybody to see this evening, and I think to a certain degree it surprised Chelsea, especially when everything seemed to be conspiring against us at the end of normal time.
"We had a back four that was thrown together to a certain extent, there were a lot of guys out there who haven't played together and there were a number who haven't had a great deal of football neither."
- 01 Butland
- 02 Bardsley Dismissed after an earlier booking
- 17 Shawcross
- 26 Wollscheid
- 05 Muniesa (Wilson - 49' Booked )
- 06 Whelan
- 16 Adam
- 18 Diouf
- 14 Afellay (Shaqiri - 76' )
- 10 Arnautovic
- 19 Walters (Odemwingie - 90' )
- 07 Ireland
- 09 Odemwingie
- 12 Wilson
- 21 Sidwell
- 22 Shaqiri
- 24 Given
- 25 Crouch
- 01 Begovic
- 05 Zouma
- 24 Cahill
- 26 Terry
- 06 Baba Booked (Nunes do Nascimento - 70' )
- 07 Ramires (Traore - 80' )
- 12 Mikel
- 22 Willian
- 08 Oscar
- 10 Hazard
- 19 Diego Costa (Remy - 33' )
- 14 Traore
- 15 Djilobodji
- 16 Nunes do Nascimento
- 18 Remy
- 28 Azpilicueta
- 32 Amelia
- 36 Loftus-Cheek
Ref: Kevin Friend