Spurs took a 1-0 lead into the match following the first leg at Wembley earlier this month, but that advantage was wiped out inside 45 minutes as goals from N'Golo Kante and Eden Hazard deservedly gave Chelsea the upper hand at half time.
Fernando Llorente pulled a goal back to level things up on aggregate once again five minutes into the second half, though, and despite both sides creating chances for a winner it came down to penalties to decide the London derby.
Missed spot kicks from Eric Dier and Lucas Moura proved costly for the visitors as another chance of silverware slipped through their fingers, setting up Chelsea for a showdown with holders Manchester City in the final at Wembley on February 24.
The match began at a really good tempo with plenty of intensity from both sides, but chances were at a premium in the opening exchanges and Pedro saw the only real sight of goal blocked by Toby Alderweireld before it could test Paulo Gazzaniga.
Chelsea's new number nine Gonzalo Higuain watched on from the stands as Olivier Giroud initially toiled up front, while Spurs clearly missed the attacking trio of Harry Kane, Dele Alli and Son Heung-min.
Hazard was causing problems after being challenged to improve by Maurizio Sarri, though, and it was his cross which forced the corner from which Chelsea broke the deadlock shortly before the half-hour mark.
Spurs could only clear the ball as far as Kante on the edge of the box, and his first-time effort flew through three pairs of legs - including Gazzaniga's on the line - before nestling into the back of the net as Chelsea levelled things up on aggregate.
Things went from bad to worse for Spurs soon after when Ben Davies limped off with a groin injury, and Chelsea almost added a second when Hazard reached the byline before playing a low cross into the box which Giroud could not smuggle in at the front post.
The home side did take the aggregate lead just two minutes after that, though, as Pedro and Cesar Azpilicueta combined to find Hazard, who steered the ball home for his first goal in seven games.
Alderweireld survived a late penalty shout as Spurs reached half time without suffering any further damage, although they went into the break having failed to register a single shot during the opening 45 minutes.
It looked as though it could be a similar story in the second half when Chelsea started quickly again, Jorginho drawing a save from Gazzaniga after Spurs had once again left players unmarked at the edge of the box from a corner.
Spurs did soon register their first shot when Alderweireld fired a well-struck volley over the crossbar, and their first shot on target provided their goal as the hitherto anonymous Llorente levelled things up on aggregate with a stooping header from Danny Rose's cross.
Suddenly penalties loomed, although Chelsea had the chance to take that option back off the table immediately after Tottenham's goal when Giroud was played through, only to be denied by Gazzaniga when he should have done better.
A string of yellow cards followed - including one for Chelsea boss Sarri - as the match continued to heat up in the chilly conditions, and chances arrived just as frequently as the bookings during a high-tempo second half.
Hazard should have done better when he was released by Kante following a failed interception from Moussa Sissoko, but the Belgian blasted wide of the near post having opted to take the shot early.
Tottenham created a golden opportunity of their own shortly after the hour mark when Christian Eriksen produced an exquisite pass into the path of Llorente, but the Spurs goalscorer tried to take his shot on first time and completely missed the ball when he had the time to take a touch.
Chelsea once again responded as Hazard's dazzling footwork almost created another clear opening, although the danger did not end there for Spurs and they needed Gazzaniga to deny Pedro from the follow-up.
Mauricio Pochettino turned to Lucas off the bench in an attempt to find a winner inside normal time, and the Brazilian almost made an immediate impact when he was picked out by a huge goal kick forward by Gazzaniga, only to fire his finish into the side-netting.
It was Chelsea who finished the match stronger, though, and Hazard sent one 25-yard effort bobbling narrowly wide of the far post with Gazzaniga beaten.
Giroud had two glorious chances to nick a winner in stoppage time, but the Frenchman took an airshot with his first before glancing a wicked Emerson Palmieri cross wide of the far post with the last meaningful action of the 90.
Both teams scored their first two spot kicks in the shootout before Dier blazed his over the crossbar to hand Chelsea the advantage, and Kepa Arrizabalaga's save to deny Lucas then allowed David Luiz to seal the victory for Sarri's side.
The Blues will now take on holders Man City in the final next month as Sarri bids for the first major trophy of his managerial career, whereas Spurs are again made to wait to end their 11-year trophy drought.
CHELSEA (4-3-3): Kepa; Azpilicueta, Rudiger, Luiz, Emerson; Kante, Jorginho, Barkley (Kovacic 81'); Pedro (Willian 76'), Giroud, Hazard
TOTTENHAM (4-4-2): Gazzaniga; Aurier, Alderweireld, Vertonghen, Davies (Rose 33'); Dier, Winks, Sissoko (Sanchez 80'), Eriksen; Lamela, Llorente (Lucas 68')