England captain Kane broke the deadlock in London and Pierre-Emile Hojbjerg doubled the lead late on as Spurs moved to within one point of leaders Arsenal.
There were chances for both sides in the first half but it remained 0-0 at the break after Everton did well to hold off a late surge from the home side.
The turning point came just before the hour mark when referee Paul Tierney ruled that Kane had been brought down by Everton goalkeeper Jordan Pickford and pointed to the spot.
Kane fired his effort into the bottom-left corner for his ninth Premier League goal of another prolific season.
Hojbjerg completed a 2-0 win in the 86th minute, condemning Frank Lampard’s side to their second consecutive league defeat.
Tottenham are flying, level on 23 points with second-placed Manchester City and just one point behind north London rivals Arsenal, with City and the Gunners both in action on Sunday.
Antonio Conte’s team have now won all five of their home league games this season.
Conte is hoping Richarlison will not be sidelined for long after the Brazilian limped off during Tottenham’s 2-0 win over Everton.
The loss of Richarlison against his former club to a calf injury early in the second half is a concern for Conte amid a hectic club schedule and for the player just a month out from the World Cup.
“He will have an MRI to understand it tomorrow, we hope it is not serious,” said Conte.
“We have to be really happy because in my opinion we played a good game, we created a lot of chances and had injuries during the game.
“That could have created something negative but we found a right solution to the game.”
Not for the first time this season, Spurs were far from a slick attacking force.
However, Kane’s form is covering any cracks as the England captain took his tally to 10 goals for the season.
He then played a part in setting up the second goal four minutes from time as the ball eventually broke for Hojbjerg and the Dane’s deflected effort found the top corner.
Everton arrived at the Tottenham Hotspur Stadium with the joint best defensive record in the Premier League and their new-found solidity under Frank Lampard ensured there would be no repeat of the 5-0 thrashing the Toffees suffered when the sides last met in March.
However, scoring goals has been the issue for Lampard’s men and the lack of threat from the visitors meant there was little doubt over the outcome after Kane had smashed home from the spot just short of the hour mark.
“The penalty is the difference in the game,” said Lampard. “I thought until then everything we were doing was good.”
Jordan Pickford had been largely untroubled in a tepid first 45 minutes from the hosts, but Tottenham cranked up the pressure after the break.
Pickford beat away a volley from his England teammate Kane before Son Heung-min headed tamely into the arms of the Everton keeper.
However, Pickford was to blame for the opening goal as he spilt Matt Doherty’s shot and then brought down Kane as he tried to collect the loose ball.
Kane had uncharacteristically fired over one of two penalties he took in Tottenham’s 3-2 win over Eintracht Frankfurt in midweek.
But normal service was resumed as he drilled the ball into the bottom corner despite Pickford going the right way.
Everton were forced to come out of their shell in the final 15 minutes as Lampard threw on Dominic Calvert-Lewin in search of an equaliser.
Yet, that only offered Spurs more space on the counter-attack and when Kane fed Rodrigo Bentancur down the right, the Uruguayan picked out Hojbjerg, who used the run of Son to distract the Everton defence before firing past Pickford.
In the early kick-off, Leicester edged off the bottom of the Premier League despite a goalless draw with Crystal Palace.
Leicester’s only league win this season was a 4-0 victory against Nottingham Forest earlier this month but that proved to be a false dawn for the 2016 champions.
The Foxes, who have the leakiest defence in the division, were solid at the back against Palace at the King Power but failed to capitalise on their chances.
“The players are fighting and running and giving everything,” said Leicester boss Brendan Rodgers.
“I always felt after the summer market we would have to fight our way through the season but the players are doing that and hopefully we will get better results as the season goes on.”
Leicester signed defender Wout Faes from French club Reims in the closing hours of the transfer window to replace Chelsea-bound defender Wesley Fofana in their only significant piece of business.
Nottingham Forest ‒ back in the Premier League for the first time in 23 years ‒ had the chance to leapfrog Leicester but lost 1-0 at Wolves.
Steve Cooper’s men have taken just one point from their past seven games after losing out in a tale of two penalties at Molineux.
Ruben Neves scored from the spot to move managerless Wolves out of the bottom three before Jose Sa saved Brennan Johnson’s penalty at the other end.
Fulham and Bournemouth remain in the top half of the table after sharing the points in a 2-2 draw at Craven Cottage.
Bournemouth extended their unbeaten run to six games under caretaker boss Gary O’Neil, but will rue missing out on three points after leading twice at Fulham.
Dominic Solanke slotted home the opener from Philip Billing’s cut-back in the second minute.
Issa Diop levelled midway through the first half with a towering header from Andreas Pereira’s corner, but Bournemouth retook the lead just seven minutes later when Solanke teed up Jefferson Lerma.
Aleksandar Mitrovic had been a doubt due to a foot injury, but Fulham’s talisman made his mark from the penalty spot to salvage a point after he had been hauled down inside the area by Lerma.
The heavyweight clash of the weekend takes place on Sunday when champions Manchester City face Liverpool at Anfield.
Leaders Arsenal are away to Leeds while Chelsea play at Aston Villa and Manchester United host Newcastle.