Maurizio Sarri got the desired reaction, having questioned the mentality of his Chelsea players in the aftermath of the weekend's defeat to Arsenal, as N'Golo Kante and Eden Hazard overturned the 1-0 first-leg deficit before half-time.
Spurs, 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 for Fernando Llorente to net an aggregate equaliser.
The shootout 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 fire the decisive kick home 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. Luckily, the Frenchman was not required to take one as the Blues sailed through after Dier and Lucas Moura's misses.