Chelsea will face Manchester City in the Carabao Cup final after beating Tottenham 4-2 on penalties at Stamford Bridge.
Goals from N'Golo Kante and Eden Hazard had seen Chelsea overturn Spurs' first-leg lead inside 38 minutes, but Fernando Llorente's header ensured the semi-final second leg ended 2-2 on aggregate.
With neither side able to find a winner, spot-kicks were needed and Eric Dier and Lucas Moura missed for Spurs, allowing David Luiz to fire the winner from 12 yards.
The dramatic victory means Chelsea will take on holders City in the final on February 24 at Wembley, where Maurizio Sarri has the chance to win the first trophy of his managerial career.
Sarri demanded a response from his players after Saturday's below-par performance at Arsenal, and that was exactly what he got as Chelsea responded to their manager's dressing down to turn the tie on its head inside 45 minutes.
Chelsea: Kepa (7), Azpilicueta (6), Luiz (7), Rudiger (6), Emerson (6), Jorginho (6), Kante (7), Barkley (6), Pedro (6), Giroud (7), Hazard (7).
Subs: Kovacic (5), Willian (6).
Tottenham: Gazzaniga (6), Aurier (6), Alderweireld (7), Vertonghen (6), Davies (5), Dier (5), Winks (6), Sissoko (6), Eriksen (6), Lamela (6), Llorente (6).
Subs: Rose (6), Sanchez (5), Lucas (5).
Man of the Match: Eden Hazard
Spurs' first-leg advantage lasted less than half an hour as a cleared corner fell for Kante on the edge of the area, and his scuffed shot fizzed past Paulo Gazzaniga, courtesy of a deflection off Moussa Sissoko.
Tottenham added Ben Davies to their mounting list of injuries on 33 minutes and Hazard soon made matters worse, applying the finishing touch to a slick Chelsea attack to put the hosts ahead on aggregate for the first time.
Hazard looked to add a third in first-half stoppage-time but Toby Alderweireld was deemed to have made enough contact with the ball while challenging his compatriot from behind, preventing a certain goal.
Within five minutes of the restart, though, Spurs were back on level terms as Llorente outmuscled David Luiz and Antonio Rudiger to reach a brilliant Danny Rose cross and powered a header into the back of the net.
There were three Chelsea changes following the defeat at Arsenal as Emerson Palmeiri, Ross Barkley and Olivier Giroud were recalled. Spurs made five changes from the win at Fulham as Moussa Sissoko, Paulo Gazzaniga, Serge Aurier, Ben Davies and Eric Dier returned.
That goal gave Spurs a timely shot in the area and ignited the cup tie, with tempers flaring on 56 minutes after Kante dived in on Rose in front of the baying home crowd.
What started off as a passive game was now frenetic. Hazard should have restored Chelsea's aggregate lead before the hour when he raced in behind Serge Aurier, but the Belgian's drive swerved into the side-netting.
Llorente then spurned a gilt-edged chance to double his tally when Christian Eriksen's lofted pass played him through on goal, the Spaniard shanking a shot wide when he had time and space to bring the ball under control.
Chelsea continued to pile on the pressure as Hazard waltzed into the area and chances fell for Olivier Giroud and Pedro, but the Spurs defence stood firm.
Mauricio Pochettino brought Moura off the bench on 67 minutes and it nearly proved to be an inspired substitution as the Brazilian latched onto Gazzaniga's long pass, but he could only fire wide at the near post.
Giroud had the chance to send Chelsea to the final in the third minute of stoppage-time but he glanced a header wide as the tie went to penalties.
Eriksen, Willian, Erik Lamela and Cesar Azpilicueta all scored before Dier blazed over. Jorginho moved Chelsea closer to victory with an unorthodox penalty before Kepa Arrizabalaga saved Moura's effort to give Chelsea match point.
And up stepped Luiz, who hammered home the crucial spot-kick to send Chelsea to Wembley as Spurs slipped to a fifth defeat in their last six semi-final appearances.
- Chelsea have reached the League Cup final for the eighth time; the joint-most final appearances by a London club in the competition (also Spurs and Arsenal).
- Maurizio Sarri is the sixth manager to take Chelsea to a League Cup final and just the fourth to do it in his first season in charge (after Vialli, Mourinho and Grant).
- Chelsea won just their second League Cup penalty shoot-out in their last six attempts, with both shootout successes coming at home.
- This was the third time that Spurs have failed to progress from a League Cup semi-final having won the first leg, alongside crashing out against Newcastle in 1975-76 and Arsenal in 1986-87.
- In all competitions, Chelsea have lost none of the 46 games in which Eden Hazard has scored in at Stamford Bridge (P46 W39 D7 L0).
- Since the start of last season, Chelsea's Eden Hazard has been directly involved in seven goals in his eight League Cup games (4 goals, 3 assists).
- N'Golo Kante has scored four goals in 30 apps for Chelsea in all competitions this season - as many as he netted in his previous three seasons in England combined (4 in 129 apps).
- Fernando Llorente has scored eight goals in his last five starts for Tottenham in all competitions.
Maurizio Sarri: "The players reacted really well [top my criticism]. I didn't attack my players, I only said we had a problem. We need this motivation to continue now. I was very happy with the performance before the penalties. If there had been a different outcome I would have still been happy."
Mauricio Pochettino: "Disappointing to lose in that way, but it can happen. We fought, were brave and played really well. Until we conceded the game was under control. I can only feel proud given all the circumstances and can only congratulate my players, that's the way we wanted to play and show we can compete at this kind of level."
Both clubs resume their FA Cup campaigns on Sunday. Chelsea host Sheffield Wednesday (6pm) while Tottenham travel to Crystal Palace (4pm) in their respective fourth-round ties.