Spurs struggled to work themselves into the lead, only earning their breakthrough from a Jordan Pickford error as he brought down Kane for the 60th minute spot-kick after Frank Lampard's Everton missed good opportunities in the first half.
"We had to be patient," Kane, who has nine league goals this season, told Sky Sports. "The spaces opened up as they got tired; thankfully we kept knocking on the door and got a couple of chances. It was a really good win."
Antonio Conte made three changes from the side which beat Eintracht Frankfurt 3-2 on Tuesday, but their first half performance against Everton was flat and Spurs struggled to get a sight at their opponents' goal.
It was Everton who forced the best chances in the first 45 minutes. Demarai Gray was the first recipient, beating Rodrigo Bentancur to a ball over the top and racing through on goal but he could only fire over from close range.
And just before half-time, Amadou Onana raced onto goal, but his effort was also skewed over.
In the 60th minute, though, Spurs found the breakthrough as Kane dispatched from the penalty spot.
Pickford was adjudged to have fouled Kane after clawing for a parried ball in the box and the England striker, on his 400th appearance for Spurs, fired past the Everton goalkeeper despite both opting for the same corner.
"Harry's [Kane] smart," Lampard told Sky Sports. "And when I say that, I'm not criticising him. I'm just saying good players understand the moments where they get the touch and take contact and that was an example of that."
Lampard made substitutions to try to gain a foothold but failed to truly trouble Spurs in the closing stages.
And Conte's side made sure of victory four minutes from time as Hojbjerg struck a late blow. Midfield partner Bentancur found Hojbjerg in the box and the Danish midfielder sent a deflected shot past Pickford to wrap up all three points for Spurs.