Pompey forward Reeco Hackett-Fairchild had the best chance of the first-half following a clever flick-on from strike partner Colby Bishop. With the ball dipping, the 24-year-old hit a volley on the turn, forcing Fraser Forster into a good save.
Such openings were simply not there for Spurs. Heung-min Son and Bryan Gil saw plenty of the ball but struggled to carve through a well-organised defence, a unit who appeared to be galvanised by the recent sacking of the Cowley brothers at Fratton Park.
As much credit as the visitors deserve for their defending, this is still a League One team to have conceded over a goal per game this season and with only two clean sheets in their last six outings.
Then came the second-half. As is so often the case with Spurs, they emerged a different team after the break, with Emerson Royal hitting the post with a brilliant header minutes before Kane’s fantastic opener.
The England captain picked up the ball outside the box, injecting some life into an otherwise fairly pedestrian Tottenham attack, before getting it back from Ryan Sessegnon and curling a shot past Josh Griffiths.
Oliver Skipp should have increased the lead but somehow blazed over from close range, keeping the prospect of an unwelcome replay alive even despite Spurs’ control on the game before the home side saw the fixture out.