We were indebted to Petr Cech, who made a string of fine saves, but the veteran goalkeeper could do nothing to prevent the Tottenham striker scoring the only goal of the game.
The match got off to a typically-frantic start, with the hosts determined to impose themselves on the game while we were happy to sit back and soak up the early pressure.
With Jack Wilshere snapping into tackles and stroking the ball about in the centre of the park, we soon settled into the game and created the first chance of any note.
It came as Wilshere collected the ball outside the area and bisected the Tottenham defence with a delightful reverse pass to set Pierre-Emerick Aubameyang clear on goal - but the flag was raised.
It was a desperately tight call, but despite it going against us we continued to push on although the hosts also looked dangerous in possession.
Indeed, they wasted a glorious chance to open the scoring when the lively Heung-Min Son picked out Kane in the area, but the striker headed over from close range.
Just before the break Hector Bellerin went close with a shot from outside the area that flew over the bar, but the two sides couldn’t be separated at the interval.
But it was the hosts who started the second period the brighter, and they took full advantage when Kane rose highest in the area to head home a cross from the left.
The Tottenham forward has a fine record against us, and he came close to doubling his side’s lead minutes later but glanced a header inches wide of the far post.
The hosts had their tails up, Kane in particular and he sent a stinging volley at goal but Cech was equal to it as we fought to stay in the game - and the Arsenal keeper made a superb stop soon after to palm away a Christian Eriksen free kick.
Alexandre Lacazette and Alex Iwobi were thrown into the mix as we looked to change the shape of the game, and the next chance was ours as Wilshere sent a curling shot at goal but Hugo Lloris was able to make the save.
Son, Dele Alli and Erik Lamela all missed good chances in quick succession as we left space open as we tried to claw our way back into the game.
Cech was certainly the busier of the two goalkeepers and he was called upon again with 15 minutes remaining to make a fine save to deny Kieran Trippier from close range.
Try as we might, we could not find a way back - although Lacazette had a late chance that he put over the bar - and we left the national stadium empty-handed for the first time since 2011.