Spurs were the better team from the off but struggled to break down the Toffees in the first half, with Richarlison going off with an injury. However, the hosts came back stronger after the break, with Kane winning a penalty after visiting goalkeeper Jordan Pickford fouled him inside the box.
The Spurs hitman made no mistake from 12 yards, slamming the ball into the bottom corner beyond Pickford's despairing dive to break the deadlock.
In the closing stages, Tottenham's persistence paid off as Hojbjerg doubled their advantage to seal the contest.
With 23 points in ten games, Antonio Conte's men remain in third place in the league standings, while Everton, after back-to-back defeats, have slumped to 14th.
On that note, here are the player ratings for Spurs:
Hugo Lloris - 6.5/10
The Tottenham custodian didn't have to make a single save on the night, thanks to Everton's wayward finishing.
Cristian Romero - 7/10
He impeded Everton's movements a few times with timely clearances and tackles, and was strong in the air.
Eric Dier - 6.5/10
He helped keep Neal Maupay under wraps, demonstrating excellent positional intelligence and marking. Dier pushed his team forward with some good long balls.
Ben Davies - 7/10
The Welshman drove forward with poise and purpose and added an edge to Tottenham's attacks. However, his propensity to play on the offensive left acres of space in behind for Everton to exploit.
Matt Doherty - 6.5/10
He was relatively quiet in the opening stanza but grew into the game after the break and looked dangerous on the right side.
Pierre-Emile Hojbjerg - 7/10
Miscommunication between him and Romero gifted Amadou Onana a great chance, but the midfielder fired his effort over the bar, much to Hojbjerg's relief.
The Dane later made up for that with a late goal that put the tie to bed.
Rodrigo Bentancur - 7.5/10
The Uruguayan set up Hojbjerg's goal with a fabulous cross and was generally superb throughout.
Ivan Perisic - 7.5/10
He was an important figure in Tottenham's attack with his incisive runs and inventive crosses. Perisic completed four of eight crosses and registered four key passes.
Richarlison - N/A
Richarlison was very active early on and seemed likely to score against his former team. However, he had to go off in the first half with an injury in what was an early blow for Tottenham.
Heung-min Son - 5/10
Son's erratic campaign continued with another disappointing showing just days after his two-goal heroics in the UEFA Champions League. He struggled to find space to work into while his crosses were wayward too.
Harry Kane - 8/10
Tottenham's biggest threat, Kane kept Everton's defenders busy and won a penalty early into the second half which he converted to put his team in front.
Ratings of Tottenham substitutes against Everton
Yves Bissouma (52' for Richarlison) - 7/10
His arrival marked a huge boost for Spurs' midfield, which looked more robust and steely after his arrival.
Djed Spence (90+3' for Doherty) - N/A
He was among the stoppage-time substitutions Conte made to see the game off.
Oliver Skipp (90+4' for Bentancur) - N/A
It was too late for him to make any impact.
Lucas Moura (90+4' for Kane) - N/A
He came on at the same time as Skipp when the game was almost up.