Spurs surged to a thrilling 3-1 win in the most recent Premier League meeting between the sides but Wembley played host to a more cagey contest on this occasion.
Kepa Arrizabalaga's ill-advised challenge on Kane allowed Tottenham to claim their slender advantage ahead of the return clash, as the England captain converted from 12 yards with typical aplomb.
N'Golo Kante hit the post for Chelsea, who controlled large spells of the second-half only to gain little change out of a miserly Spurs backline.
Callum Hudson-Odoi brought a vibrancy to the Chelsea attack and forced Paulo Gazzaniga into a routine save before life became more complicated for the Spurs goalkeeper's opposite number.
Kepa hared off his line as Kane chased Toby Alderweireld's raking ball and, once a VAR referral confirmed the England captain was onside, a penalty and booking were inevitable.
Chelsea's number one guessed correctly but could not redeem himself as Kane's spot-kick beat him for pace.
The post twice came to Tottenham's aid late in the first half, with Kante side-footing Marcos Alonso's cross against Gazzaniga's right-hand upright before Hudson-Odoi's deflected cross also bounced to safety.
Kane was unable to get the better of Kepa from further afield in the 52nd-minute – the Spain international making a superb stop from his 25-yard piledriver.
Kante tested Gazzaniga from similar range before Blues defender Andreas Christensen missed a glorious chance when Ross Barkley flicked on Eden Hazard's corner.
Tottenham's back-up goalkeeper dangerously fumbled an 86th-minute cross from Hazard, while a crunching stoppage-time challenge from Danny Rose left the full-back with an apparent hamstring injury as Mauricio Pochettino's men held out.
Advantage Spurs but all to play for
Despite Pochettino's talk of prioritising the Premier League and the Champions League, this competition and the prospect of a major honour clearly matters a great deal to him and his players. That much was clear from the team-sheet, with Kane able to score a first goal in this competition since 2014. The weight of the occasion also weighed on a Spurs team who showed none of the freewheeling destructiveness that saw them take Chelsea apart two months ago. It was a stifled affair, particularly after half-time, but Tottenham got the job (half) done.
Hudson-Odoi shines as Bayern rumours swirl
Widespread reports of being the subject of a £35million bid from Bundesligagiants Bayern Munich might be expected to weigh heavily on an 18-year-old winger. But Hudson-Odoi romped around Wembley without a care in the world, his direct play bringing pace and purpose at odds with a sometimes dreary contest. He looks made for this stage.
Jorginho comes up short against Tottenham again
The sight of Jorginho flailing in pursuit of Son Heung-min was a defining image of Chelsea's defeat in the league encounter on this ground. Although there were no causes for viral embarrassment this time around, Chelsea's deep-lying playmaker was unable to exert his usual control on the contest, even when the visitors were in the ascendancy after half-time.
