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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 Spurs to claim their slender advantage ahead of the return clash, as the England captain converted from 12 yards with typical aplomb, albeit after consultation with the video assistant referee (VAR).
N'Golo Kante hit the post for Chelsea, which 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.