Spurs overturned Chelsea's 2-1 first leg lead and also ended their 26 game hoodoo over their London rivals.
Jimmy Floyd Hasselbaink was also sent off, albeit in a case of mistaken identity as it was Melchiot who actually deserved to be punished for slapping Teddy Sheringham around the face.
Sheringham had just put Spurs 3-0 up after strikes by Steffen Iversen, after just one minute, and Tim Sherwood had seized the initiative.
And there was no way back for Chelsea as Simon Davies and substitute Rebrov rounded off the scoring to give Spurs a final spot against Blackburn in February.
Iversen took just 105 seconds to make his mark and level the two-legged affair.
Sheringham's dummy left Taricco in the clear but when his shot was parried by keeper Cudicini, Terry delayed a straightforward clearance.
That enabled Iversen to bundle the ball home from close range and there were soon further problems for Chelsea as Zenden went off injured yet again.
Blues boss, Claudio Ranieri's solution was to bring on Sam dalla Bona in an attempt to stem Tottenham's continuing pressure.
Then, when Poyet's half-volley was tipped over, it was Sherwood who struck again from the ensuing corner on 32 minutes.
Anderton's delivery looked to have been mishit but the ball flew straight to his midfield colleague Sherwood, who hit an unstoppable drive into the top corner.
Early in the second half Chelsea's hopes of reaching the final were all but over.
Within four minutes of the restart, Anderton delivered a cross from the left and Poyet chested the ball into the path of Sheringham.
Sheringham struck a first-time volley into the far corner as the Chelsea defence stood motionless.
Chelsea's growing frustration boiled over and Hasselbaink, and not Melchiot, was sent off with 35 minutes left by referee Mark Halsey.
Zola and Mikael Forssell both came on but Spurs soon struck again.
Davies beat keeper Cudicini at the near post and then, with four minutes left, Rebrov converted a chance from close range to complete Chelsea's misery. Chelsea: Carlo Cudicini, Marcel Desailly, William Gallas, John Terry, Mario Melchiot (Gianfranco Zola, 70), Emmanuel Petit (Mikael Forssell, 66), Mario Stanic, Boudewijn Zenden (Sam Dalla Bona, 11), Frank Lampard, Jimmy Floyd Hasselbaink, Eidur Gudjohnsen
A minute later Mikael Forssell scored what was little more than a consolation goal for a bitterly disappointing Chelsea side.
Tottenham: Neil Sullivan, Chris Perry, Ledley King, Anthony Gardner, Simon Davies, Gustavo Poyet (Oyvind Leonhardsen, 83), Tim Sherwood, Darren Anderton, Mauricio Taricco, Teddy Sheringham, Steffen Iversen (Sergei Rebrov, 83)
Subs: Gavin Kelly, Sergei Rebrov, Oyvind Leonhardsen, Matthew Etherington, Alton Thelwell
Subs: Ed de Goey, Albert Ferrer, Sam Dalla Bona, Gianfranco Zola, Mikael Forssell
Referee: M Halsey
Attendance: 36,100