A low-key first half had featured almost nothing in the way of goalmouth incident, with Arsenal wasting several promising breaks with poor final passes.
But Spurs fashioned its best chance when Kane had a header stopped by Peter Cech, while Eric Dier failed to connect properly with a loose ball inside the area.
The home side needed only four minutes of the second half to take the lead, Kane leaping superbly to meet a fine Ben Davies cross and steer his header wide of Cech.
Two minutes later it could have been two, Kane again meeting a cross but this time heading wide from close range.
The Spurs striker, seemingly on a one-man mission to see off Arsenal, thumped a volley straight at Cech with 55 minutes gone, and the visiting keeper tipped away a Christian Eriksen free kick soon afterwards.
After 64 minutes, Arsenal made a double change with Arsene Wenger bringing on Alex Iwobi and Alexandre Lacazette for Henrikh Mkhitaryan and Mohamed Elneny.
Wenger's side fashioned a chance after 67 minutes as Jack Wilshere made space but saw his curling effort from the edge of the box saved, but within moments Spurs should have been two up only for Dele Alli to prod past the advancing Cech but wide.
Cech was straight back into the action, saving well from substitute Erik Lamela, who had replaced Son Heung-Min, and the keeper excelled again as he beat away a Kieran Trippier volley to keep the Gunners in the game.
Arsenal tried to muster some late pressure, but Pierre-Emerick Aubameyang was thwarted by the offside flag as he tried to race through and Lacazette sent an angled shot agonisingly wide two minutes into time added on.