The Blues wasted no time to hand new signing Edouard Mendy his debut in goal, while Spurs were without Heung Min-Son and Dele Alli but gave Sergio Reguilon his first start.
Spurs were given an early set-back 19 minutes in after Timo Werner moved out of position in the area to pick up Cesar Azpilicueta's cross from the right to apply a near post finish for the opening goal.
Desperate defending from the Blues at the other end denied Gedson Fernandes the equaliser after Kurt Zouma helped out his goalkeeper Mendy with a strong defensive clearance.
Tougher dues would come Mendy's way seconds from the half-hour mark as Erik Lamela's strike towards the bottom corner forced the Frenchman to use his legs to deny his Argentine counterpart.
Spurs persisted to draw level despite trailing their visitors at the break, as Mendy turned Reguilon's first-time effort from the left soon following halftime behind for a corner-kick.
Attention would turn to the substitutes bench for the home side on 62 minutes with Fernandes sacrificed for Danish international Pierre-Emile Hojbjerg before Kane was introduced eight minutes later.
The closing quarter had openings either end as Reguilon steered wide after heading Serge Aurier's cross in off the ground with Werner denied a brace from close-range by Hugo Lloris.
Spurs drew the cat out of the bag six minutes from conclusion after Lamela volleyed home Reguilon's looping delivery from the left with calls for an offside waved away.
The match went to a penalty shoot-out to determine a victor, and Spurs would successfully win the shoot-out to book a quarter-final berth, with Harry Kane netting the decisive kick before Mason Mount missed for the visitors.