Kane netted from close range during the second half but it was a mixed afternoon for the forward after he was stretchered off with a badly-twisted ankle, and it could keep him on the sidelines for the foreseeable future.
After being handed his first start in a Spurs shirt, Moussa Sissoko almost gave his team the lead as early as the fourth minute when he lashed a shot marginally wide of the far post before Kane saw a close-range effort well saved by Jordan Pickford.
The young Black Cats stopper again came to his team's rescue with a save from Mousa Dembele down to his left, while Toby Alderweireld was also denied by the England Under-21 international after finding the target with a header.
Sunderland had offered nothing in the final third but they almost took a surprise lead when Jermain Defoe latched on to a long ball from Pickford before seeing his effort at the near post saved by Hugo Lloris.
After a brief lull in play, the first half ended with both teams coming agonisingly close to breaking the deadlock with Son Heung-min hitting the post for Spurs before Sunderland's Steven Pienaar had a close-range attempt cleared off the line by Kyle Walker.
Spurs returned for the restart keen to re-stamp their authority on proceedings and they created another opportunity through Son, who sent the ball into the side-netting when he perhaps should have passed the ball across goal.
However, just before the hour mark, the home side finally had their goal through Kane, who was perfectly placed to take advantage of indecision in the Sunderland defence to find the bottom corner from inside the six-yard area.
While Spurs initially failed to build on the goal, Sunderland were offering nothing going forward and their opponents were able to ease their way into the closing stages with their lead still intact.
With four minutes left, Spurs were dealt a significant blow when Kane twisted his ankle in a challenge with a defender, and he could find himself sidelined for the club's upcoming games after being stretchered down the tunnel.
Erik Lamela was also denied a goal by an outstanding save from Pickford, but it mattered little as Spurs held on to continue their unbeaten start to the campaign.