Kane broke the Toffees’ resistance from the penalty spot after being tripped in the area by England team-mate Jordan Pickford, who had failed to hold on to Matt Doherty’s long-range effort.
Pierre-Emile Hojbjerg added a second late on with a curling strike that took a slight deflection off Alex Iwobi on its way in.
Spurs dominated the early stages but the two best chances of the first half fell to the visitors, who will be disappointed not to have put away both opportunities.
Demarai Gray failed to hit the target after racing on to Conor Coady’s long ball over the top of the Tottenham defence, before Amadou Onana blazed over the crossbar following Hojbjerg’s poor touch.
Kane sent a rasping volley straight at Pickford early in the second half but there was nothing the Toffees keeper could do about the 29-year-old’s spot-kick – his 14th goal in 15 Premier League appearances against Everton – or Hojbjerg’s late finish.
The victory lifts Conte’s team level on points with second-placed Manchester City and just a point behind north London rivals Arsenal, while Everton drop to 14th.
The victory lifts Spurs on to 23 points from 10 league matches – their highest tally at this stage of a Premier League season – while Kane has now scored in five successive top-flight games for the first time in his career.
It was a win the home side thoroughly deserved after a dominant display, although they rode their luck in the first half and could have found themselves behind at half-time.
But Frank Lampard’s team were no match for Spurs in the second half and it wasn’t long before Spurs’ pressure told, Pickford bringing down England team-mate Kane inside the six-yard box with an outstretched arm.
Kane’s penalty lifts him to within eight goals Jimmy Greaves’ all-time Tottenham record of 266.
Hojbjerg secured the win with five minutes remaining, collecting Rodrigo Bentancur’s delivery before finding the far corner via a deflection off Iwobi.
The only sour note from the hosts’ perspective was a second-half injury to former Everton forward Richarlison, who went down clutching his left calf and was replaced by Yves Bissouma.