Tottenham Hotspur grabbed an 89th-minute winner against Crystal Palace to move into fourth place after a determined home side had seemed set for a battling draw at Selhurst Park.
Kane produced a fine header to break Palace's resistance and notch the 150th club goal of his career after an afternoon of missed chances had appeared certain to cost Spurs points.
The visitors had an early scare when they failed to deal with a Palace corner and the ball ricocheted around before Dele All blasted it away.
But from that moment, Mauricio Pochettino's side took control in the spring-like sunshine at Selhurst Park and confusion in the home defence allowed Ben Davies to steer the ball across goal before it was scrambled away.
After 11 minutes, Spurs should have been in front when a mistake by Patrick van Aanholt let in Harry Kane, whose close-range strike was well blocked by Palace keeper Wayne Hennessey.
Van Aanholt was close to making amends soon afterwards when his dangerous cross was dealt with by Hugo Lloris as Christian Benteke waited, and then an early Spurs ball forward just eluded the onrushing Alli.
Spurs thought they had a penalty on the half-hour when Kane was brought down in the box by Timothy Fosu-Mensah, only for an offside flag to rescue the home side.
But Palace got more of a foothold in proceedings as the half wore on and, with the break approaching, Andros Townsend tried his luck with an ambitious effort when he had teammates waiting in the middle.
Spurs again dominated possession at the start of the second half, but an indication of their frustration at failing to break through came when, with 49 minutes gone, Kane hammered a wild shot into the stand.
Within minutes, two good chances had come and gone as Christian Eriksen thumped over and then Kane slashed wide from close in.
Serge Aurier brought a good save from Hennessey and then, with 66 minutes gone, Spurs boss Mauricio Pochettino brought on Lucas Moura for Erik Lamela.
As the game entered its final 20 minutes, Palace's Alexander Sorloth was denied space inside the area by a good Aurier tackle, and then Alli went down as Hennessey came to meet him but the referee was not interested in penalty claims.
With 73 minutes gone Spurs were frustrated once more as Hennessey produced another good stop from a low Davies shot that had seemed destined to find the bottom corner.
Hennessey was at it again as Kane's low cross-shot took a deflection on its way through after good work from substitute Son Heung-Min, and then Aurier somehow failed to get a telling touch onto a fierce low Davies ball into the area.
Kane's low strike with four minutes of normal time remaining fizzed just wide of the far post - but, as so often, he had the last word from Eriksen's cross.