Kane was handed the winner on a plate in the 59th minute after Papy Djilobodji completely missed the ball in the six-yard box as he tried to clear his lines but the England striker went off in the 87th after turning his ankle making a tackle.
"Maybe he has some problem with his ligaments but today it is difficult to assess. We need to wait until tomorrow for the scan," manager Mauricio Pochettino told a news conference after his side moved up to third in the standings on 11 points.
Only an inspired performance by Sunderland keeper Jordan Pickford kept the hosts at bay as Tottenham managed 19 goal attempts before the break, including an effort from their South Korea forward Son Heung-min that hit the post.
In the second half, Son hit the side netting and rolled another effort across the goal before Kane, who scored his first league goal of the season at Stoke City last week, struck by pouncing on Djilobodji's mistake after Dele Alli's knockdown.
Spurs continued to press, with Eric Lamela twice going close and Vincent Janssen blazing over in added time. Sunderland finished with 10 men after Adnan Januzaj was sent off for a foot-up challenge on Ben Davies, having already been booked.
(Reporting by Neil Robinson; Editing by Ken Ferris)
