Son Heung-min added a third early in the second period with the pick of the goals, making the most of some terrible Chelsea defending in doing so, and though substitute Olivier Giroud bagged a late consolation, the Blues' unbeaten start to the season ended in their 19th match under Maurizio Sarri.
Tottenham had lost four of their previous seven league meetings against the division's top-six elite here ahead of this match, but they were quick out of the blocks in this latest tussle against a side that headed into the contest a point above them.
Alli's sixth goal in five matches against Chelsea gave Tottenham something to build on, and they did exactly that as Kane tucked a long-range shot out of the reach of Kepa, who expected David Luiz to block the ball rather than move out of the way.
Son had forced Kepa into a save down low prior to that vital second goal, though it may well have been a different story at 1-0 had Eden Hazard been awarded a penalty after being caught in the box by Juan Foyth - referee Martin Atkinson feeling that there was not enough contact to justify pointing to the spot.
Stunned by Tottenham's second goal, Chelsea simply could not sustain any pressure and again relied on Kepa to keep out Toby Alderweireld and Son inside the box, pulling off fine saves to deny both players.
The Blues' best opening fell the way of star performer Hazard, but his 20-yarder was easy enough for Hugo Lloris to deal with.
Spurs put the game out of Chelsea's sight 10 minutes into the second period as, after Willian blazed over the bar at one end, Son showed the Brazilian how it is done by carrying the ball 50 yards - dodging weak challenges from Jorginho and Luiz in the process - before bending it past Kepa.
N'Golo Kante wasted a chance to drag his side back into the match when guiding a Ross Barkley cross wide of the target, but Tottenham could easily have extended their lead as the chances kept falling their way.
Kane fired over when picked out by Serge Aurier six yards from goal and Alli's curler clipped the top of the net, with yet more poor Chelsea defending on display for both opportunities.
The final goal of the match would go Chelsea's way, however, as Giroud came off the bench to nod a Cesar Azpilicueta delivery across Lloris, although it was not enough to prevent a first league defeat in 13 matches for Sarri and his men.
TOTTENHAM HOTSPUR (4-2-3-1): Lloris; Aurier, Alderweireld, Foyth, Davies; Sissoko, Dier; Eriksen, Ali, Son (Lamela 78'); Kane
CHELSEA (4-3-3): Kepa; Azpilicueta, Rudiger, Luiz, Alonso; Jorginho, Kante, Kovacic (Barkley 58'); Willian (Giroud 78'), Morata (Pedro 58'), Hazard