Kane's second goal of the season came to Spurs' rescue at White Hart Lane but manager Mauricio Pochettino will be sweating on the striker's fitness after he had to be carried off in the 89th minute.
The victory sends Tottenham up to third in the table and maintains their unbeaten start, while Sunderland remain without a win in the Premier League this season.
The Black Cats were reduced to 10 men late on when Adnan Januzaj was shown a second yellow card but the sending off barely affected the contest, which Spurs dominated from start to finish.
The home side's afternoon, however, was marred by injury concerns with Mousa Dembele and Eric Dier both taken off in the second half, before Kane was also forced to depart.
Tottenham managed 19 attempts at goal in the first period to their opponents' three and for the first 25 minutes in particular, Sunderland's contribution was solely in blocking, deflecting and finally hoping that their goalkeeper Jordan Pickford would keep the hosts at bay.
He succeeded too, twice denying Kane from point-blank range and then holding long-range efforts from Dembele and Moussa Sissoko.
Sissoko, booed by the visiting supporters for his past allegiances with Newcastle, was bright on his full debut and along with Son Heung-min, the pair caused Sunderland problems with their pace and direct running.
Son went closest with a fizzing shot that struck the outside of the post and soon after applied the first of three quick-fire Spurs efforts, his blocked by Kane, Sissoko's volley by Lee Cattermole before Victor Wanyama dragged the last effort wide.
Amidst their frantic but committed defending, Sunderland arguably enjoyed the best two chances of the half.
Jermain Defoe latched on to Pickford's punt down field but was unable to steer past Hugo Lloris, and on the stroke of half-time Steven Pienaar had the goal at his mercy six yards out, but somehow picked out Kyle Walker on the line.
Son continued to terrorise his opposite number Jason Denayer after the interval, unleashing into the side-netting and flashing a ball across goal, but when the goal came, it was fellow Sunderland defender Papy Djilobodji who was culpable.
Dele Alli nodded down Walker's cross and Djilobodji should have made a routine clearance but instead sliced it into the path of Kane, who was perfectly placed to stab home.
Erik Lamela, on for the hobbling Dembele in the latter stages, had two chances to put Tottenham out of sight and Son looked to have a strong case for a penalty when he was scythed down by Djilobodji late on.
Kane's afternoon took a turn for the worse when he appeared to injure his ankle while lunging for a tackle and the striker left the field on a stretcher, replaced by summer signing Vincent Janssen.
Januzaj picked up his second yellow card in the 89th minute for a late challenge on Ben Davies and that was the final nail in the coffin for Sunderland, as Spurs held on for a narrow victory.
Source : PA
Source: PA