Newcastle, spurred on by a buoyant atmosphere in their first game since being taken over by a Saudi Arabia-led consortium, flew out of the traps and went ahead after just 108 seconds when Callum Wilson headed in from a Javi Manquillo cross.
But the bright start soon fizzled out as Spurs hit back in the 17th minute through Tanguy Ndombele, who lashed home from the edge of the box after being played in by Sergio Reguilon.
Things got worse for the hosts five minutes later when Pierre-Emile Hojbjerg played an excellent ball over the Newcastle defence and Kane beat the offside trap to smartly dink it over Karl Darlow.
"We got beaten by the better team after a wonderful start. Defensively, we had the same problems we've had for a while... we got our noses in front but did not defend well as a team," said Newcastle boss Steve Bruce, who took charge of his 1000th game as a manger.
Spurs coach Nuno Espirito Santo told Sky: We knew what was coming, with everything going on around Newcastle, in the end it was a positive performance. We stayed in the game, we kept it steady. The moment we started moving it around, we started causing them problems."
Play was suspended for nearly 25 minutes just before halftime after a fan collapsed in the East Stand of the ground. Newcastle said that the supporter's condition had been stabilised and they were on their way to hospital.
Son Heung-min added a third from close range in first-half stoppage time before Newcastle were reduced to 10 men in the 83rd minute when Jonjo Shelvey saw red for a second booking. An own-goal from Eric Dier set up a tense finish but Spurs held on.
"We needed to score next (in the second half) to give ourselves a chance but we never had any sustained pressure over them... but we still stuck and stayed with them," added Bruce.
Spurs moved up to fifth in the standings with 15 points from eight games. Newcastle are second from bottom with three points.