England captain Kane provided the spark after a goalless first half at Selhurst Park by heading the opener on 48 minutes and following up with a well-taken second five minutes later.
Four goals were shared at Elland Road as Leeds and West Ham played out a 2-2 draw that increased the pressure on Hammers manager David Moyes.
Rodrigo secured a point for the hosts with a 70th-minute equaliser after West Ham had come from behind to lead with goals either side of half-time from Lucas Paqueta – a penalty after Jarrod Bowen was fouled by Pascal Struijk – and Gianluca Scamacca.
Wilfried Gnonto had opened the scoring for Leeds in the 27th minute.
The Hammers are now outside the bottom three only on goal difference after Nottingham Forest climbed out of the relegation zone with a 1-0 win at bottom side Southampton.
Taiwo Awoniyi scored the only goal, turning the ball home on 27 minutes from a Brennan Johnson pass after Lyanco gave the ball away.
The result – Forest’s first away win since returning to the top flight – left Saints two points adrift at the foot of the table with their new manager Nathan Jones still to claim a point after four matches in charge.
It also meant Everton, thrashed 4-1 by Brighton on Tuesday, fell into the drop zone.
Danny Ings equalised 12 minutes from time as Aston Villa, with World Cup-winning goalkeeper Emiliano Martinez back in their side, grabbed a 1-1 draw in their derby clash against Wolves at Villa Park.
Wolves, 19th in the table, had been on course for a much-needed win after Daniel Podence put them ahead on 12 minutes.