Maurizio Sarri got the desired reaction, having questioned the mentality of his Chelsea players in the aftermath of Saturday's defeat to Arsenal, as N'Golo Kante and Eden Hazard overturned the 1-0 first-leg deficit before half-time.
Tottenham, without any of Harry Kane, Dele Alli or Son Heung-min in a competitive match for the first time since October 2016, found a response of their own by pegging Chelsea back with Fernando Llorente netting an aggregate equaliser.
The shoot-out opened in nerveless fashion, but Eric Dier blazed over the bar and Lucas Moura shot tamely to Kepa Arrizabalaga's left, leaving David Luiz to steer in and spark raucous celebrations.
Returning Tottenham midfielder Dier clattering into David Luiz before fouling Jorginho set a scrappy early tone prior to the hosts going in front after 27 minutes.
Kante was given ample room to lash a partially cleared corner back towards goal and his shot zipped through the legs of Ben Davies, Moussa Sissoko and Spurs goalkeeper Paulo Gazzaniga.
Hazard was revelling in the freedom of having Olivier Giroud playing alongside him as a specialist target man and he fed Pedro to find the overlapping Cesar Azpilicueta, before continuing his run to coolly convert the Spain full-back's cross.
Danny Rose came on for the injured Davies in between the Chelsea goals and the England left-back played a key role as an improved Spurs levelled five minutes into the second half, with Llorente heading home his delivery from the left.
Llorente was unable to find the clean contact Christian Eriksen's delicious swerving cross deserved – he made way for Lucas in the 68th minute – while Pedro fired too close to Gazzaniga at the end of a frantic Chelsea attack.
Giroud, with Gonzalo Higuain looking on from the stands, passed up a pair of glorious chances in stoppage time and penalties were required for the Blues to book a date at Wembley.
Sarri tough love ploy pays off
Bearing in mind the Stamford Bridge dressing room's tendency to chew up and spit out head coaches, the wisdom of Sarri calling out first his team as a whole and then his star player over recent days appeared ill-advised. But his XI responded with the sort of sharp, tenacious display before half-time that Sarri had wondered aloud might be beyond them, while Hazard showed plenty of his dazzling best. Tottenham's reserves of battling qualities under Mauricio Pochettino, even in the face of a punishing injury list and a lack of reinforcements, should not be questioned any time soon.
Jorginho gets Chelsea on the front foot
Sarri's unswerving commitment to his deep-lying playmaker has invited growing criticism of late, meaning Jorginho's sharpest performance since being ransacked by these opponents in November was particularly timely. The ex-Napoli man's expertise in position is not up for debate, but his passing came at a more agreeable tempo, while no Chelsea player gained possession more often. It was a display capped with a delightfully cheeky kick under shoot-out pressure.
